THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

CTOMETRY 
GIFT  OF 


Dr.  Hugh  ?.   Brown 


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Ophthalmic  Lenses 

DIOPTRIC  FORMULA 
FOR  COMBINED  CYLINDRICAL  LENSES 


THE  PRiSM-DIOPTRY 

AND    OTHER 

OPTICAL  PAPERS 


WITH    ONE    HUNDRED   AND   TEN    ORIGINAL    DIAGRAMS 


Charles  F.  Prentice,  M.  E. 

"OPTICIST" 


(Second  Edition) 


PUBLISHED   BY 

THE  KEYSTONE  PUBLISHING  CO. 

809-81 1-813  North  19TH  Street,  Philadelphia,  U.S./ 

1907 

All  rights  reserved 


Copyright,  1900,  by  B.  Thorpe 
Publishes,  op  The  Keystone 


Add'l 
GIFT 


iqoy 


PREFACE 


In  the  Treatise  on  Ophthalmic  Lenses  an  endeavor  is  made,  by- 
means  of  graphic  and  analytic  methods,  to  guide  the  novice  upon  a  path 
by  which  he  may  easily  acquire  a  knowledge  of  lenticular  refraction  with- 
out recourse  to  mathematics.  The  text  is,  therefore,  confined  to  the 
description  of  a  series  of  diagrams,  by  aid  of  which  the  principles  involved 
are  presented  in  their  natural  order  of  succession. 

Shortly  after  publication  of  the  above  treatise,  Dr.  Burnett,  of 
Washington,  D.  C. ,  kindly  suggested  the  production  of  plastic  models  of 
combined  cylindrical  lenses,  by  placing  an  incomplete  set  of  models 
of  his  own  conception  with  the  author  for  further  elaboration.  With  it 
came  the  request,  if  possible,  also  to  produce  two  combinations  in  which 
the  cylinders  were  to  be  united  at  angles  other  than  right  angles.  As  a 
result  of  the  author's  research,  during  the  time  devoted  to  the  construc- 
tion of  the  latter  more  especially,  and  with  a  view  to  establish  confi- 
dence in  the  precision  of  these  models,  a  complete  mathematical  demon- 
stration of  the  refraction  by  combined  cylindrical  lenses  was  first  presented . 
in  1888. 

The  purpose  of  republishing  the  Dioptric  Formula?,  in  Section  II,  is 
to  present  the  important  results  then  obtained,  with  as  much  abridgment 
of  the  calculations  as  permissible. 

While  the  diagrams  have  been  prepared  with  great  care,  yet  they  are 
somewhat  at  variance  with  the  laws  of  true  perspective,  ov*-ing  to  the 
author's  desire  to  strictly  preserve  therein  all  important  circles  and  right 
angles  referred  to  in  the  text. 

293 


Among  the  many  attempts  to  solve  this  problem,  this  is  the  only  one 
whose  formuliE  contain  exclusively  the  known  quantities,  namely,  the 
angle  between  the  cylinders  and  their  foci  or  powers.  Furthermore,  this 
is  the  only  solution  which  has  ever  disclosed  the  sixteen  laws  inherent  in 
a  pair  of  superposed  cylindrical  lenses — a  fact  which  may  be  easily  veri- 
fied by  comparing  this  solution  with  those  of  Bonders,  Reusch,  Heath, 
Jackson,  Hay,  Weiland,  Suter  and  Thompson,  whose  formulae  are  also 
iar  less  simple. 

The  main  object  in  republishing  the  Prism-Dioptry  and  other  original 
papers,  in  Section  HI,  is  to  present  them  collectively  to  the  student  of 
optometry,  who  in  most  instances  will  now  find  it  very  difficult  to  gain 
access  to  the  numbers  of  the  journals  in  which  the  various  original  articles 
have  appeared  during  the  past  ten  years.  This  action  seems  to  be  further 
justified  by  the  frequent  inquiries  made  for  reprints,  which  have  long  since 
been  exhausted,  and  also  for  the  reason  that  American  manufacturers 
have,  since  1894,  universally  adopted  the  prism-dioptral  system  for  their 
entire  marketable  product  of  prisms.  This  should  also  prove  an  incentive 
for  all  who  make  use  of  such  prisms  to  become  thoroughly  familiar  with 
the  character  and  capabilities  of  the  prism-dioptry. 

The  manuscripts  on  prisms  were  generally  approved,  before  original 
publication,  by  Dr.  Burnett,  to  whom  the  author  is  indebted  for  having 
called  attention  to  the  necessity  for  a  new  system  of  numbering  prisms, 
and,  indeed,  also  for  having  suggested  the  name  of  its  unit,  the  prism- 
dioptry.  Therefore  its  original  form  of  spelling  is  herein  retained.  For 
the  sake  of  greater  clearness  it  has  also  been  deemed  advisable  to  make 
several  important  additions  to  the  original  text,  which,  in  its  present  form, 
may  at  least  be  considered  authentic  with  reference  to  modern  ophthalmic 
prisms. 

The  appended  revised  papers  will,  it  is  hoped,  prove  of  permanent 

interest,  as  they  contain  features  of  scientific  value  not  to  be  found  in  any 

]-.r.nd-book  or  treatise  on  optics.  ^  -.^    ^ 

Charles  F.   Prentice 


GENERAL  CONTENTS 


SECTION  I 

OPHTHALMIC  LENSES ii 

SECTION  II 

DIOPTRIC  FORMUL/E  FOR  COMBINED  CYLINDRICAL  LENSES  .   .     49 

SECTION  III 

THE   PRISM-DIOPTRY  AND  OTHER  OPTICAL   PAPERS 99 


CONTENTS 


OPHTHALMIC  LENSES 
SECTION  I 


PAGE 

Refraction    13 

Prisms 16 

Simple  Lenses 20 

Compound  Lenses. 

1.  Congeneric  Meridians  (convex) 29 

2.  "                 "           (concave) 37 

3.  *  Contra-Generic  "          (convex  and  concave) 39 

ToRic  Lenses. 

Congeneric  and  Contra-Generic  Meridians 44 

Table  of  Dioitral  Numeral^ 47 

Tables  of  Crossed  Cylinders. 

1.  Congeneric  Meridians ^  -> 

2.  Contra-Generic  Meridians S 

*  Sec  Diopiric  Formula;  for  Combined  Cylindrical  Lenses. 


CONTENTS 


DIOPTRIC  FORMUL/E 
FOR  COMBINED  CYLINDRICAL  LENSES 

SECTION  II 

PAOK 

I.  Dioptric  Formula  for  Combined  Congeneric  Cylindrical  Lenses. 

1.  Relative  Positions  of  the  Primary  and  Secondary  Planes  of  Refraction   ■  •  •  53 

2.  Positions  of  the  Primary  and  Secondary  Focal  Planes 59 

3.  Relations  between  the  Primary  and  Secondary  Focal  Planes 62 

11.  Dioptric  Formul,«  for  Combined  Contra-Generic  Cylindrical  Lenses. 

1.  Relative  Positions  of  the  Principal  Positive  and  Negative  Planes  of  Refraction  69 

2.  Positions  of  the  Positive  and  Negative  Focal  Planes 72 

3.  Relations  between  the  Positive  and  Negative  Focal  Planes 75 

in.  Dioptral  FoRMUL/ii:  FOR  Combined  Cylindrical  Lenses. 

I.   Relation  between  the  Principal   Planes  of    Refraction   and  the   Refractive 

Powers  of  the  Cylinders     81 

IV.  Sphero-Cyltndrical  Equivalence     87 

V.  Verification  of  the  Formulae 95 

1.  Tables  in  Verification  of  the  Dioptric  Formulse 97 

2.  Tables  in  Verification  of  the  Dioptral  Formulae 97 

LIST  OF  PLATES 

FRONTISPIECE 

Half-Tone  Plate  of  Dr.  Swan  M.  Burnett's  Models,  Demonstrative  of  Cylindrical 
Refraction,  constructed  by  the  author  in  accordance  with  the  Formulas  for  Congeneric 
Cylindrical  Lenses. 

PLATES  I  AND  II 

The  Refraction  by  Combined  Congeneric  Cylindrical  Lenses,  demonstrated  in  three 
diagrams.      Plate  I  on  page  52.      Plate  II  on  page  So. 

PLATES  III  AND  IV 

The  Refraction  by  Combined  Contra  generic  Cylindrical  Lenses,  demonstrated  in 
three  diagrams.    Plate  III  on  page  68.     Plate  IV  on  page  86. 


CONTENTS 


THE  PRISM-DIOPTRY 

AND    OTHER   OPTICAL    PAPERS 

SECTION  III 

PAGB 

The  PRisM-Droi»TKV    99 

A  Metric  System  of  Numbering  and  Measuring  Prisms 105 

1.  The  Relation  of  the  Prism-Dioptry  to  the  Meter  Angle Ill 

2.  The  Relation  of  the  Prism-Dioptry  to  the  Lens-Dioptry II5 

The  Perfected  Prismometer 125 

The  Prismometric  Scale 139 

On  THE  Practical  Execution  OF  Ophthalmic  Prescriptions  Involving  Prisms  •  •  151 

A  Problem  in  Cemented  Bi-Focal  Lenses,  Solved  by  the  Prism  Dioptry 157 

Why  Strong  Contra-Generic  Lenses  of  Equal  Power  Fail  to  Neutralize  Each 

Other. i6i 

The  Advantages  of  the  Sphero-Toric  Lens 167 

The  Iris,  as  Diaphragm  and  Photostat 173 

The  Typoscope 183 

The  Correction  of  Depli-tted  Dynamic  Refraction  (PRtiiKYOPiA) 185 


SECTION  I 


OPHTHALMIC  LENSES 


THEIR    REFRACTION    AND    DIOPTRAL    FORMUL/C 


WITH   THIRTY-SIX  ORIGINAL  DIAGRAMS 


REFRACTION. 


§  1.  The  change  wrought  in  the  direction  of  oblique  rays  of  light,  on 
their  passage  from  one  transparent  medium  to  another  of  different  density, 
is  called  Refraction.  As  our  proposed  treatment  of  its  manifestation  by 
lenses  is  to  be  strictly  elementary,  we  must  first  define  the  law  of  refraction, 
at  least  so  far  as  applied  to  parallel  rays,  in  air,  impinging  upon  and  pass- 
ing through  transparent  optical  glass.  In  the  accompanying  diagram,  Fig. 
1,  a  piece  of  glass  of  considerable  thickness,  having  parallel  surfaces  a-& 
and  c-d,  is  presented  as  an  isolated  vertical  section  a-h-c-d,  which  is  ex- 
posed to  the  oblique  ray  i  in  the  same  plane  of  the  surrounding  air. 


S?|      p"!        A 


For  convenience  we  shall  term  the  ray  prior  to  its  contact  with  the  glass, 
the  incident  ray,  i;  the  ray  during  transit  within  the  glass,  the  refracted 
ray,  e*  e^  ;*  and  the  refracted  ray  after  exit,  the  final  ray,  f. 


*  The  use  of  superior  indices  wiU  not  prove  conflicting  j>s  algebraic  Taluee  are  excluded. 

13 


14 


REFRACTION. 


§  2.  Refraction  manifests  itself  by  an  acute  bend  in  the  direction  of  an 
oblique  ray  of  light,  i,  at  the  point  of  incidence,  e^,  in  passing  from  one 
conducting  mediimi  to  another,  a-b-c-d,  of  different  density.  Hence,  a 
ray  passing  from  one  into  and  through  another  medium  is  bent  both  at  the 
point  of  entrance  e^  and  of  exit  c". 


/''^^y- 


Fig.  1. 


By  virtue  of  the  deflection  or  bend  alluded  to,  the  incident  ray,  (',  must 
include  a  different  angle,  a,  with  the  perpendicular  p^  from  that,  Z^,  of  the 
refracted  ray  e^  e" ;  and  it  is  by  the  trigonometrical  values  s  and  s^  of  these 
angles,  which  have  been  found  to  bear  a  constant  proportion  to  each  other, 
that  we  are  enabled  to  give  expression  to  the  amount  of  deflection  sustained 
by  a  ray  in  passing  from  one  medium  to  another. 


§  3.  Experiment  has  shoMoi  that  the  proportion  *r  remains  a  constant 
value  for  any  obliquity  of  a  ray  incident  to  the  same  medium,  and  yet, 
that  it  possesses  a  different  value  by  substituting  one  medium  for  another. 

It  has  therefore  been  considered  expedient  to  establish  the  value  of  -^ 
for  all  transparent  media,  in  the  specific  case  of  a  ray  passing  from  air  into 
them ;  such  values  being  known  as  the  refractive  indices  of  the  substances. 

To  illustrate  the  graphical  method  by  whicli  we  may  arrive  at  the  di- 
rection of  the  refracted  ray,  when  tlio  index  of  refraction  and  the  direc- 
tion of  the  iucident  ray  are  known,   we   shall   select  the   index   for   crown 


KKFRACTTON-.  15 

glass  =  l.o,  bv  introtliicin*;  the  proportion  ^,  =^  ^  =  1.5  in  the  construc- 
tion as  follows : 

After  erecting  the  perpendicular  p^,  take  from  a  scale  of  equal  parts  the 
value  for  s  =  ."),  and  transfer  it  between  c'  and  h^  beneath  the  ray  i,  upon 
the  line  c^  b. 

In  the  same  manner  transfer  the  value  for  s^  —  2  from  e^  to  a^,  upon  the 
line  e^  a,  and  in  both  of  these  points,  &^  and  a^,  erect  perpendiculars.  The 
perpendicular  at  ¥  will  intersect  the  ray,  i,  at  a  point,  v,  which  limits  the 
radius  of  a  circle  drawn  from  c'  as  a  center ;  and  by  the  circle's  intersection 
with  the  perpendicular  at  a}  the  point,  x,  defining  the  direction  of  the  ray, 
e^  e-,  is  fixed. 


§  4.  As  a  ray  of  light  is  propagated  backwards  or  forwards  on  the  same 
path,  the  index  of  refraction  from  a  denser  medium  into  air  is  the  inverse 
proportion  from  that  of  air  into  the  medium,  hence  ~  is  the  proportion  by 
which  the  direction  of  the  final  ray,  f,  is  to  be  determined  when  the  direc- 
tion of  the  ray,  e^  e",  is  known. 

We  therefore  erect  at  e-  the  perpendicular  p^  and  transfer  the  value  of  s^ 
=  2  beneath  the  ray  e^  e-  from  e-  upon  the  line  e~  d;  likewise  the  value 
for  s  =  3  from  e-  upon  the  line  e-  c,  and  erect,  as  before,  in  the  points  d^ 
and  c^  the  perpendiculars. 

The  perpendicular  at  d'^  will  intersect  the  ray,  e^  e-,  at  a  point,  y,  limit- 
ing the  radius  of  a  circle  from  the  point  e^;  the  point,  z,  at  the  circle's 
intersection  with  the  perpendicular  in  r'^  establishing  the  direction  of  the 
final  ray,  /. 

As  p^  and  p-  are  parallel  from  the  construction  it  follows  that  the  ray  /  is 
parallel  to  i,  and  therefore  of  the  same  direction. 


PRISMS. 

§  5.  Pursuant  to  the  spirit  of  our  intention  to  avoid  mathematical  for- 
mulaB,  we  shall  seek  to  arrive  at  a  conclusion  respecting  the  deflection  in- 
curred by  a  ray  in  passing  through  a  medium  with  oblique  plane  surfaces, 
confining  ourselves  as  before  to  isolated  vertical  sections. 


Fig.  2. 


Fig.  3. 


Specifically,  we  shall  select  two  right-angled  prisms  of  var3'ing  angles, 
«*  and  a',  with  the  rays  t*  and  r  incident  perpendiculary  to  the  vertical 
sides  a'h^  and  a^b^,  so  as  to  avoid  refraction  on  the  incident  sides,  as  shown 
in  the  vertical  sections,  Fig.  2  and  Fig.  3,  respectively.  At  e-  and  e^  the 
rays  i'  and  i^  suffer  refraction  in  the  proportion  -  •  =-|,  according  to  §4, 
and  which,  if  carried  out  in  the  construction,  as  before  indicated,  deter- 
mines the  directions  of  /-  and  /"',  respectively,  as  shown. 

In  the  future  we  shall  have  occasion  to  refer  to  the  line  dv,  which  is  the 
perpendicular  from  v  upon  a  line  coincident  with  the  ray  i^  when  the  latter 
is  parallel  to  the  base  b^c^  of  the  prism.  Fig.  3.  Under  such  circumstances 
the  displacement  dv  of  the  final  ray  P  is  associated  with  a  mathematical 
dependency  upon  the  angle,  a^,  of  the  prism,  and  the  index  of  refraction 
-JL.     See  page  108. 

16 


PRISMS.  17 

From  tliG  construction  it  follows  that  the  ilnal  ray  f^  (Fig.  3)  intersects 
the  horizontal  line  /<°  at  n^;  and  /-  (Fig.  2)  at  a  more  distant  point,  n-,  not 
shown.  By  a  comparison  of  the  prismatic  section  Fig.  2  with  Fig.  3,  we 
observe  that  by  a  decrease  of  the  angle  from  a^  to  a-  the  perpendicular  p^ 
has  a  greater  tendency  to  parallelism  with  the  horizontal  line  /i°  than  />*. 
8uch  parallelism  being  realized — when  a-  c^  is  parallel  to  (r  h-  or  a^  =  0° 
— would  result  in  the  value  s^  vanishing  in  the  incident  ray  i~,  and  s  in  the 
final  ray  f-,  by  virtue  of  the  decrease  of  the  angles  of  incidence  and  refrac- 
tion in  the  proportion  3  to  3,  thus  establishing  the  coincidence  of  the  inci- 
dent and  final  rays,  and  placing  the  point  (n-)  of  intersection  at  infinity 
respecting  the  horizontal  line  Zi". 


§  6.  In  general  we  may  therefore  be  permitted  to  assume  that  the  greater 
the  angle,  a,  of  obliquity  of  the  surfaces  (Fig.  4)  the  greater  will  be  the  de- 
flection of  the  final  ray  f,  and  the  closer  to  the  base  he  of  the  prism  will  be 
its  intersection  n  with  the  horizontal  line  h^.  In  looking  through  a  prism 
the  refraction  manifests  itself  by  an  apparent  change  from  the  true  position 
of  an  object  0,  to  that  of  its  image  OS  when  viewed  from  the  point  n.  The 
author's  suggestion  that  this  phenomenon  should  form  the  basis  of  compari- 
son in  measuring  prisms  was  adopted  by  American  manufacturers  in  1895. 
The  unit  of  prismatic  refraction  is  equal  to  a  deflection  (dv.  Fig.  3)  of  one 
centimeter  at  one  meter's  distance,  and  is  called  the  prism-dioptry. 


§  7.  By  confining  our  observations  to  the  relative  directions  of  the  inci- 
dent and  final  rays,  we  may  easily  memorize  the  law  of  refraction,  for  a 
medium  included  within  plane  surfaces,  in  the  following  manner: 


18 


PKISMS. 


1,  a.  The  direction  of  a  ray  remains  unchanged  in  passing  through 

opposite  parallel  surfaces  of  a  transparent  medium,  or 
b.  The  incident  ray  i  and  the  final  ray  /  are  parallel  when  the  former 
is  projected  obliquely  upon  a  transparent  medium  included  within 
parallel  surfaces. 

2,  a.  The  direction  of  a  ray  is  changed  in  passing  through  opposite 

oblique  surfaces,  by  a  deflection  of  the  final  ray  /  toward  the  region 
of  their  gi'eatest  distance  apart,  or 

b.  The  incident  ray  i  and  the  final  ray  /  are  obliqvie  when  the  former 
impinges  upon  a  transparent  medium  included  within  oblique 
surfaces,  or 

c.  The  apex  of  the  angle  formed  by  an  obliquity  of  the  incident  and 
final  rays  is  always  directed  toward  the  apex  of  the  angle  of 
obliquity  of  the  surfaces. 

The  law  of  refraction  (2)  finds  its  graphical  demonstration  in  the  follow- 
ing figures,  wherein  we  have  introduced  the  medium  glass  as  being  inter- 
sected by  imaginary  vertical  and  horizontal  planes,  V  and  H,  coordinate  at 
ihe  point  of  exit  e-  for  the  final  ray  /. 


Fig.  5. 


Fig.  6. 


Prism,  Base  vertical;  Refraction 
horizontal. 


Prism,  Base  horizontal;  Refraction 
vertical. 


§  8.  The  Figures  5  and  6  are  of  particular  interest  to  us,  as  they  illustrate 
a  very  vital  element  in  our  future  consideration  of  the  refraction  by  cylin- 
drical lenses,  namely,  that  the  refraction  is  strictly  confined  to  the  plane 
whose  intersection  with  the  mediimi  defines  the  obliquity  of  its  surfaces. 
Thus,  for  an  obliquity  of  the  surfaces  in  the  horizontal  plane  H  (Fig.  5), 


PRISMS. 


19 


■we  find  tho  refraction  active  in  the  horizontal  plane  (i"  to  p),  and  for  an 
obliquity  of  the  surfaces  in  the  vertical  plane  1'  (Fig.  (>)?  the  refraction  is 
active  in  the  vertical  plane  (i^  to  f^). 

Here,  in  the  sense  that  the  final  rays  are  confined  to  the  plane  of  inci- 
dence, we  may  term  the  refraction  passive  in  respect  to  its  right-angled 
coordinate  plane.  Thus  in  Fig.  5  the  refraction  is  passive  with  regard  to 
the  vertical  plane,  and  in  Fig.  0  witlr  regard  to  the  horizontal  plane. 


Fig.  7. 
Prism,  Base  oblique;  Refraction  diametrically  opposed. 


§  9.  It  is  evident  that  the  refraction  is  active  in  one  and  passive  in  the 
other  plane  for  a  medium  whose  surfaces  are  oblique  in  but  one  plane,  so 
that  to  obtain  the  refraction  active  in  both  fixed  planes  an  obliquity  of  the 
surfaces  relative  to  each  plane  would  he  necessary.  In  such  a  medium 
(Fig.  7),  if  we  consider  the  refraction  merely  with  regard  to  the  horizontal 
obliquity  of  the  surfaces,  the  final  ray  would  take  the  direction  P-Ji,  and,  if 
independently  for  the  vertical  obliquity,  the  final  ray  would  assume  the 
direction  f^-v.  Therefore,  with  due  consideration  to  the  obliquity  in  both 
planes,  the  refraction  must  include  both  properties  of  deflection  and  result 
in  a  final  ray,  /,  which  is  directed  to  a  point,  m,  defined  by  projection  of 
the  apportioned  horizontal  and  vertical  displacements,  dh  and  dv.  As  this 
is  a  prism  whose  base  is  really  set  diagonally  to  the  fixed  right-angled 
coordinate  system,  the  ray  /  must  naturally  be  refracted  in  the  direction  of 
the  greatest  distance  apart  of  the  surfaces,  through  the  point  m,  within  tht; 
diagonally  bisecting  or  oblique  plane  P. 


SIMPLE  LENSES. 

§  10.  Directing  our  attention  to  the  effect  produced  by  substituting  a 
segment  of  a  circle  for  the  line  or  c-  of  the  original  prismatic  section  (Fig. 
2),  each  succeeding  point  e~,  e^,  e*  (Fig.  8)  may  be  considered  as  one  of  a 
prism  varying  in  its  angle  a-,  a^  a*  with  that  of  its  predecessor ;  and  if  the 
construction  be  carried  out  for  each  incident  ray  i',  v',  i*  the  corresponding 
radial  lines  at  the  points,  e-,  e^,  e*  in  this  ease  substituting  the  perpen- 
dicular jr  heretofore  mentioned,  each  final  ray  /-,  /\  /*  will  be  found  to 
intersect  an  arbitrarily  selected  base  line,  /i°^  at  the  respective  points 
nr,  71^,  n*,  to  infinity. 


Fig.  8. 
Plano-convex  section. 


In  the  so-called  plano-convex  lens.  Fig.  8,  the  converging  final  rays  f,  f, 
p,  corresponding  to  the  more  central  incident  parallel  rays  i^,  i",  i''*  estab- 
lish points  n^,  n^,  n'  to  infinity,  and  possess  the  remarkable  feature  of  inter- 
secting each  other  at  a  common  point  F,  termed  the  focal  point,  which  is 
situated  upon  the  central  and  direct  ray  i-f.  According  to  §  4,  rays  ema- 
nating from  the  focal  point  F,  will  be  emitted  as  parallel  rays  r',  t®,  %'  and  i. 


*AU  future  deductions  refer  exclusively  to  such  rays. 

20 


SIMPLK    LENSES, 


21 


The  points  n*,  n^,  rr,  toward  the  leus,  correspond  to  the  more  eccentric  in- 
cident rays,  and,  in  the  sense  that  these  fail  to  assist  in  tlic  harmony  of  a 
union  of  the  final  rays  at  the  focal  point,  are  to  be  considered  a  disturbing 
element,  giving  issue  to  what  is  termed  aberration.  In  tlie  plano-concave 
lens,  Fig.  9,  the  final  rays  p,  /",  f  are  emitted  as  diverging  rays,  which 
may  be  considered  as  emanating  from  the  so-called  virtual  focal  point  F, 
situated  on  that  side  of  tlie  section  in  which  the  rays  arc  incident. 


Fig.  9. 


Plano-concave  section. 


§  11.  For  either  of  the  above  lenses  it  is  also  obvious  that  the  more  acute 
the  curvature  of  the  circle,  the  greater  proportionately  will  be  the  angles 
a^  a^,  a*,  limiting  the  obliquity  of  the  surfaces,  and  the  closer  to  the  lens 
will  be  the  focal  point  F.  Further,  as  the  curvature  of  the  circle  is  de- 
pendent upon  the  dimensions  of  the  radius,  the  latter  must  prescribe  the  dis- 
tance, D,  of  the  focal  point  from  the  medium  or  lens  whose  index  of  re- 
fraction is  known.  This  relationship  involves  mathematical  formulae  for 
which  we  refer  the  reader  to  special  treatises*  on  the  subject. 

The  greater  the  deflection  of  the  final  rays  p,  f',  f,  the  shorter  will  be 
the  distance  D,  or,  for  an  increase  in  the  refraction  we  have  a  corresponding 
decrease  of  the  focal  distance.  Hence  we  say  that  the  refractive  power 
of  a  lens  is  in  inverse  proportion  to  its  focal  distance. 


*  Elementary  Geometrical  Optics,  W.  Steadman  Aldis,  M.A.,  Cambridge,  1886.     Hand  Book  of 
Optics  for  Students  of  Ophthalmology,  W.  N.  Suter,  B.A.,  M.D.,  New  York,  1899. 


22 


SIMPLE    LENSES. 


§  12.  If  we  express  the  unit  of  refraction  by  the  numeral  1,  for  a  lens 
whose  focal  distance  D  is  equal  to  one  meter  or  100  centimeters,  lenses  of 
two.  three,  or  four  times  the  refraction  would  find  the  expression  of  their 
focal  distances  in  ^,  -J,  \,  the  focal  distance  of  the  unit,  or  50,  33J  and  25 
centimeters,  respectively. 

Tlie  unit  above  mentioned  has  been  termed  the  Dioptry,  and  is  now  the 
standard  of  refraction  in  optometrical  practice.  Values  beneath  the  unit 
are  designated  as  0.25D.,*  0.50D.,  and  0.75D,,  their  respective  focal  dis- 
tances being  four  meters  or  400  centimeters,  two  meters  or  200  centimeters, 
and  one  and  one-third  meters  or  133-J  centimeters.  Those  values  which 
are  higher  than  the  unit  are  expressed  in  whole  numbers,  including  their 
intervals  as  above.     See  page  47. 


§  13.  Assuming  the  medium  to  divide  the  aerial  space  into  negative  and 
positive  regions  (Figs.  8,  9)  as  indicated  by  the  sign  —  (minus)  on  the  in- 
cident side  of  the  mediimi,  and  the  sign  -\-  (plus)  behind  the  medium,  we 
shall  find  the  focal  point  on  the  positive  side  for  all  convex,  and  on  the 
negative  side  for  all  concave  lenses. 

In  this  sense  the  refraction  for  convex  lenses  is  considered  positive,  and 
for  concave  lenses  negative.     Hence  Fig.  8  is  +  ID.,  and  Fig.  9  is  —  ID. 


^M..........±      t 


Fig.  10. 


§  14.  By  substituting,  in  Fig.  8,  for  the  plane  side,  a  curvature  c^  con- 
centric with  C-,  the  refractive  effects  of  the  sections.  Fig.  8  and  Fig.  9,  are 
virtually  united,  as  shown  in  Fig.  10.  Owing  to  the  concave  curvature  c\ 
the  incident  ray  i  will  assume  the  direction  e^-e",  being  coincident  with  the 
focal  point  F,  which  may  also  be  practically  accepted  as  the  focal  point  for 


''D  here  being  the  abbreviation  for  Dioptry. 


SIMPLE    LENSES. 


23 


the  convex  cnrvahire  r-,  provided  tlie  thickness,  t,  of  the  medium  is  created 
infinitely  small  in  proportion  to  the  radii  r^  and  r". 

Eays  emanating  from  the  focal  point  F^  for  a  convex  curvature  c',  being 
emitted  as  parallel  rays,  §  10,  it  conditionally  follows  that  the  ray  /  will  be 
parallel  to  the  ray  i.  The  neutralization  is  the  more  complete  when  the 
curvatures  c^  and  c"  are  identical,  and  are  brought  in  contact  as  shown  in 
Fig.  11,  which,  however,  is  a  special  case. 

§  15.  Hence,  in  a  pair  of  united  convex  and  concave  sections  of  identical 
curvature,  it  follows  that  the  effect  of  the  one  is  neutralized  by  the  other, 
respecting  the  existence  of  a  focal  point  on  either  side  of  the  medium. 
This,  however,  is  strictly  only  true  for  lenses  weaker  than  9D. 

§  IG.  If  the  opposite  curvatures  be  unequal,  the  final  rays  will  unite  at 
a  focal  point  on  that  side  of  the  medium  which  corresponds  to  the  focal 
point  of  the  more  acute  curvature. 


Fig, 


Periscopic  convex  section. 
Positive  meniscus. 


Periscopic  concave  section. 
Negative  meniscus. 


Eeferring  to  the  periscopic  convex  and  concave  sections.  Fig.  12  and 
Fig.  13,  respectively,  if  we  consider  the  refraction  merely  with  respect  to 
the  front  curvature  c^,  disregarding  the  existence  of  a  terminating  back  sur- 
face, the  incident  ray  i  will  assume  the  direction  of  the  ray  e^  /S  toward  the 
focal  point  F^,  then  within  the  medium. 

Accepting  the  plane  e--p  to  be  the  limit  of  the  medium,  the  ray  e^  e^ 


24 


SIMPLE    LEJSfSES. 


would  suffer  a  second  refraction,  and  result  in  the  ray  e--f-,  directed  to  the 
focal  point  at  F-. 

To  eliminate  this  second  or  augmented  refraction,  it  would  be  necessary 
for  the  ray  e^  e^  to  impinge  upon  the  back  surface  c-  perpendicularly  at  e*. 

A  surface  effecting  this  is  obtained  by  giving  it  a  curvature  c-  prescribed 
from  the  point  F'^  as  a  center,  in  which  specific  event  the  ray  e}  <?-  traverses 
the  radius  of  the  circle,  or  the  perpendicular  at  e^  for  the  surface  c-,  thus 
fixing  the  point  F'^  as  the  focal  point  for  the  respective  periscopic  convex 
and  concave  sections. 


§  17.  Observation  of  the  figures  shows  that  the  weaker  proportionately 
reduces  the  refraction  of  the  more  acute  curvature,  so  that  the  focal  point 
F^  of  the  periscopic  is  at  a  greater  distance  from  the  medium  than  the  focal 
point  F-  of  the  plano-convex  or  concave  sections.  The  more  acute  the  cur- 
vature c^  within  the  limits  of  parallelism  with  the  eurvatiire  c\  the  more 
distant  will  be  the  focal  point  F^  from  the  medium,  so  that  the  total  refrac- 
tion for  the  respective  sections  is  equivalent  to  the  difference  of  the  appor- 
tioned numerals,  and  bears  the  sign  corresponding  to  the  more  acute  cur- 
vature &. 

Supposing,  in  a  periscopic  convex  section,  2.5D.  to  be  the  prescribed 
numeral  of  refraction  for  the  convex,  and  0.50D.  for  the  concave  side,  the 
total  refraction  will  be  2.5  —  0.50  =  2D.  convex,  or  -)-  2D. 

Similarly,  in  a  periscopic  concave  section,  2.5D.  concave  combined  with 
0.50D.  convex  equals  2.5  —  0.50  =  2D.  concave,  or  —  2D. 


Fig.  14. 
Double  or  Bi-convex  section. 


Fig.  15. 
Double  or  Bi-concave  section. 


§  18.  In  the  bi-convex  and  bi-concave  sections  Fig.  14  and  Fig.  15  it  can 
be  similarly  shown  that  the  curvature  c^  increases  the  refraction  of  cS  so 


SlMl'LE    LENSES. 


25 


that  the  total  refraction  is  expressed  by  the  sum  of  the  apportioned  numer- 
als and  bears  the  sign  associated  with  the  respective  sections. 

Thus  in  either  figure  the  numeral  for  c^  being  ID.,  and  for  c-  being 
1.5D.,  the  total  refraction  will  be  1  +  1.5  =  3.5D. 

Convex,  or  +  2.51),  for  Fig.  14,  and  concave,  or  —  2.513.  for  Fig.  15. 


Fig.  16. 


§  19.  For  a  medium  (Fig.  16)  composed  of  parallel  vertical  sections,  each 
adjacent  imaginary  section  has  its  corresponding  focal  point  at  the  same 
distance  (D)  from  the  medium,  so  that  the  refraction  for  all  central  in- 
cident parallel  rays  becomes  manifest  by  establishing  a  succession  of  these 
points,  resulting  in  the  so-called  focal  line  I  F  I. 


Fig.  17. 


Fig.  18. 


Axis  vertical;  Refraction  horizontal.  Axis  horizontal;  Refraction  vertical. 

Plano-convex  Cylindrical  Lenses. 


A  similar  succession  of  the  radial  centers  (c)  establishes  a  line  (A  cA), 
termed  the  axis  of  the  so-created  cylindrical  medium  or  lens,  which  is- 
parallel  to  the  focal  line  I  F  1,  and  in  the  same  plane. 


26 


SIMPLE    LENSES. 


§  20.  As  simple  cylindrical  lenses  have  their  surfaces  of  greatest  obliquity 
in  the  plane  wliich  is  j^erpendicular  to  the  axis,  we  here  also  find  the  re- 
fraction active  in  this  plane,  and  passive  in  the  axial  or  right-angled  co- 
ordinate plane  (see  Figures  17-20),  wherein,  as  before,  *"  and  /°  are 
associated  with  refraction  in  the  horizontal,  and  ?>  and  f^  with  refraction 
in  the  vertical  plane. 

In  a  practical  experiment  in  which  the  lens  is  held  at  some  distance 
from  the  eye,  convex  cylindrical  refraction  manifests  itself  by  an  apparent 
increase,  and  concave  cylindrical  refraction  by  an  apparent  decrease  in  the 
dimensions  of  an  observed  object  in  that  plane  which  is  at  right  angles  to 
the  axis.  In  the  axial  plane,  the  refraction  being  passive,  corresponding 
dimensions  remain  unchanged. 


Fig.  20. 


Axis  vertical;  Refraction  horizontal.       Axis  horizontal;  Refraction  vertical. 
Plano-concave  Cylindrical  Lenses. 


§  21.  To  obtain  cylindrical  refraction  of  equal  amount  in  both  planes, 
thereby  reducing  the  focal  line  to  a  focal  point,  it  would  be  necessary  to 
combine  two  identical  cylinders,  or,  to  create  a  single  lens  whose  opposite 
surfaces  are  right-angled  coordinate  cylindrical  elements  as  shown  in 
Fig.  21. 

Under  such  circumstances,  however,  the  focal  line  PFH^  for  the  front 
surface  c^  is  slightly  closer  to  the  face  of  the  lens  than  the  focal  line  l-F^2' 
for  the  back  surface  c".  Aside  from  this,  in  making  a  bi-cyliudrical  lens  it 
is  difficult  to  insure  the  chief  planes  of  refraction  being  strictly  at  right- 
angles  to  each  other,  so  that  failure  in  this  is  certain  to  increase  the  aber- 
ration. 


SIMPLE    LENSES. 


27 


§  22.  The  greater  the  distance  apart  of  the  surfaces,  c^  and  c-,  the  greater 
will  be  the  al^orrative  distance,  F^  to  F-.     Yet,  as  the  tliickncss  of  the 


Fig.  21. 
Double  or  Bi-cylindrical  Lens. 


lens  may  generally  be  accepted  as  a  vanishing  quantity  in  proportion  to 
the  focal  distance,  we  may  consider  a  common  focal  point  to  exist  for  both 
refracting  surfaces. 


Fig.  22. 
Plano-convex  Spherical  Lens. 


§  23,  Practicall}-,  however,  it  would  be  better  to  create  a  single  surface 
capable  of  producing  this  amount  of  refraction  in  the  vertical  as  well  as  in 
the  horizontal  plane.  With  this  object  in  view,  we  shall  select  the  isolated 
vertical  section  described  in  §  10,  and  cause  it  to  be  rotated  upon  the 
central  incident  and  direct  ray,  i-f,  as  its  so-called  optical  axis,  whereby 
a  plano-convex  spherical  lens  is  obtained.  See  Fig.  22.  Similar  rotation 
of  the  sections  Figs.  9,  12,  13,  14,  and  15,  would  result  in  the  so-created 
spherical  lenses  being  charact'^rized  by  the  pectioTis  employed. 


28 


SIMPLE    LENSES. 


It  is  evident  that  the  incident  and  final  rays  will  retain  their  relative 
obliquity  during  the  rotation,  so  that  all  incident  parallel  rays  have  their 
corresponding  final  rays  in  the  resulting  cone  whose  apex  is  at  the  focal 
point  F. 

To  further  illustrate,  we  may  take  advantage  of  §  9  in  its  application  to 
a  medium  having  only  one  surface  which  is  spherically  curved,  and  con- 
sequently obli<iue  in  respect  to  botli  right-angled  coordinate  planes. 


In  the  plano-convex  spherical  lens.  Fig.  33,  if  we  consider  the  refraction 
at  e^  of  the  ray  i  merely  with  regard  to  the  horizontal  refraction,  the  final 
ray  would  take  the  direction  f°-]i,  and,  if  independently  for  the  vertical 
refraction,  the  final  ray  would  assume  the  direction  f^-v.  Therefore,  with 
due  consideration  to  the  refraction  in  both  planes,  the  refracted  ray  must 
include  both  properties  of  deflection,  and  result  in  a  final  ray  f,  which  is 
directed  to  the  focal  point  F^  through  a  point  m,  of  the  oblique  plane  P, 
as  defined  by  projection  of  the  apportioned  horizontal  and  vertical  displace- 
ments dh  and  dv. 


§  24.  Finally,  we  may  therefore  conclude  that  spherical  refraction  is 
equivalent  to  the  refraction  of  right-angled  crossed  cylinders  of  identical 
curvature. 

As  in  spherical  lenses  the  refraction  is  equably  active  in  any  pair  of  dia- 
metrically-opposed meridians,  it  follows  that  both  the  lateral  and  vertical 
dimensions  of  objects  seen  through  them  will  appear  to  be  enlarged  by  con- 
vex and  diminished  by  concave  lenses,  when  these  are  held  at  some  distance 
from  the  eye. 


COMPOUND  LENSES. 


I.     CONGENERIC  MERIDIANS  (CONVEX). 


§  25.  An  asymmetrically-refracting  or  astigmatic  lens  is  one  in  which, 
the  principal  diametrically-opposed  sections  include  different  degrees  of 
refraction,  in  contradistinction  to  those  hitherto  mentioned,  in  which  imi- 
form  refraction  took  place  either  exclusively  in  one  meridian,  or  equally  in 
both  meridians. 


Fig,  24. 
Convex  Cylindro-cylindrical  Lens   (+  ci  axis  180"^ 


+  c-  axis  90°). 


Referring  to  §  21,  Fig.  21,  it  is  evident  that  the  aberrative  distance 
F'^  to  F-  may  also  be  definitely  increased  by  giving  different  amounts  of 
refraction  to  the  active  planes  or  sections  of  the  combined  cylinders.  In 
this  event  the  focal  point  ascribed  to  the  equally  curved  bi-cylindrical 
lens  will  be  effaced,  thougli  substituted  by  a  pair  of  focal  lines,  whose  dis- 
tance apart  will  be  equal  to  the  difference  between  the  focal  distances  of 
the  crossed  unequal  cylinders.  Thus  in  the  bi-cylindrical  lens,  Fig.  24, 
represented  as  consisting  of  two  crossed  convex  cylinders  (c^  and  c")  of  un- 
equal curvature,  VF'^J^  and  l-FH-  will  be  the  respective  elementary  focal 
lines.    The  distance  between  them  (F^  to  F-)  has  been  termed,  by  Sturm, 

the  "focal  interval."' 

29 


30 


COMPOUND    LENSES. 


As  the  cylinders  are  of  equal  length,  the  focal  lines  l^FH^  and  l-F-P 
would  also  be  identical  in  this  regard  if  the  apportioned  refractions  of 
the  cylinders  were  considered  independently  of  each  other. 


Fig.  24. 
Convex  Cylindro-cylindrical  Lens  (+  ci  axis  180°   o   +  c^  axis  90°). 


§  26.  The  combined  refraction  of  the  cylinders,  however,  definitely 
modifies  this  specific  condition,  and  in  the  following  manner: 

The  outermost  incident  rays  i^,  in  the  central  horizontal  plane,  which 
would  have  been  directed  to  the  points  V-  and  V  for  the  cylinder  c^,  will 
suffer  horizontal  displacement  toward  the  point  F~,  owing  to  the  activity 
of  the  refraction  in  this  plane  for  the  cylinder  c^,  and  so  establish  points 
d}  and  d^  of  the  focal  line  VFH^  for  the  combined  action  of  the  cylinders 
c^  and  c^  in  the  horizontal  plane. 

Similarly,  the  outermost  incident  rays  i^  in  the  central  vertical  plane, 
which  would  have  been  directed  to  the  points  l~  and  I-  for  the  cylinder  c-, 
will  suffer  vertical  refraction  in  this  plane  by  the  cylinder  c^  which  causes 
the  final  rays  to  cross  each  other  at  F'^  and  to  intersect  the  focal  line  l-FH'- 
at  the  points  d-  and  d-  for  the  combined  action  of  the  cylinders  c^  and  c" 
in  the  vertical  plane. 

If  we  consider  the  refraction  at  the  point  e-  of  the  circle  C  for  the  ray  l 
merely  with  regard  to  the  horizontal  refraction  of  the  surfaces,  or  the 
cylinder  c^,  the  final  ray  Avoukl  take  the  direction  e'-/?,^  intersecting  the  focal 
line  of  the  cylinder  c-  at  a  correlative  point  n- ;  but,  as  all  final  rays  for  the 
cylinder  e^  above  the  central  horizontal  plane  intersect  the  focal  line  d^F'^ 
d^,  it  follows,  through  introducing  the  cylinder  c'^,  that  the  ray  e-  h  must 


CONGENERIC    MERIDIANS, 


31 


fall  subject  to  the  influence  of  c^  for  the  combined  action  of  the  cylinders, 
thus  depressing  the  ray  e^  h  from  the  point  li  perpendicularly  to  m},  and 
consequently  also  the  point  n^  to  mr  within  the  focal  line  dr  F^  dr. 

By  an  analogous  reasoning  to  §  9  we  here  also  find  the  direction  of  the 
final  ray  /  to  be  determined  by  projection  of  the  apportioned  horizontal 
and  vertical  displacements,  dh  and  dv,  which  are  solely  dependent  upon 
the  active  meridians  of  the  cylinders  c^  and  c-. 

Increased  proximity  of  the  point  e"  to  e",  upon  the  circle  C,  will  be  as- 
sociated with  an  increased  distance  between  m}  and  F^,  and  with  an  ap- 
proach of  mr  toward  F~  for  these  points  of  intersection  of  the  final  ray  / 
with  the  respective  focal  lines  F^  d^  and  F~  d^.  The  reverse  is  evident  for 
an  advancement  of  e-  towards  e^.     (See  Fig.  25a.) 


Fig.  25a. 


§  27.  The  total  refraction  for  all  incident  parallel  rays  within  the  area 
of  the  circle  C  will  therefore  result  in  an  astigmatic  pencil  whose  focal 
lines,  d^  F'^  d^  and  d-  F^  d^,  are  limited  as  to  position  and  magnitl^de. 
This  astigmatic  pencil,  if  intercepted  at  intervals  by  a  transverse  perpen- 
dicular screen,  will  project  elliptical  areas  of  light  whose  longest  and. 
shortest  diameters  correspond  to  the  principal  meridians  of  refraction.  In 
the  immediate  vicinity  of  F^,  for  instance,  the  ellipses  have  their  longest 
diameters  horizontal ;  whereas,  in  the  vicinity  of  F~,  their  longest  diameters 
are  vertical. 

This  naturally  effects  a  reversal  of  the  ellipses,  respecting  their  diameters, 
at  some  point  within  the  focal  interval  F^-F^;  such  point  being  determined 
where  the  vertical  and  horizontal  displacements  are  alike,  and  the  section 
T,  consequently,  a  "circle  of  least  confusion." 


32 


COMPOUND    LENSES. 


§  28.  Astigmatic  refraction  in  a  lens  is,  however,  preferably  attained  by 
combining  a  spherical  with  a  cylindrical  surface,  the  requisite  conditions 
being  fulfilled  through  that  increase  or  decrease  of  the  spherical  refraction 
which  is  produced  by  and  in  the  active  meridian  of  the  cylinder. 


Fig.  25. 
Convex  Sphero-cylindrical  Lens  (+  s^  -C:  +  C^  axis  180°) — Double  Form. 


To  increase  the  refraction  of  a  positive  or  negative  spherical  lens  in  one 
meridian,  we  may  add  to  it  the  active  meridian  of  a  cylinder  bearing  the 
same  sign ;  and  to  decrease  it  in  the  same  meridian  we  may  combine  it  with 
the  active  meridian  of  a  cylinder  bearing  the  opposite  sign. 

(1)  The  combination  of  a  positive  spherical  with  a  positive  cylindrical 
surface  would  result  in  the  section  of  greatest  refraction  being  double  con- 
vex; and, 

(2)  The  combination  of  a  positive  spherical  with  a  weaker  or  less  acutely 
curved  negative  cylindrical  surface  would  result  in  the  section  of  least 
refraction  being  periscopic  convex. 

Where  the  aforesaid  combinations  are  spoken  of,  we  shall  for  conven- 
ience, apply  to  them  the  terms  double  and  periscopic  form,  respectively. 


§  29.  As  the  combination  of  crossed  convex  cylinders  of  unequal  cur- 
vatures gave  rise  to  a  pair  of  focal  lines,  to  the  novice  it  may  appear 
requisite  that  a  focal  point  and  a  focal  line  should  exist  for  the  combina- 
tion of  a  spherical  with  a  cylindrical  surface.  We  shall  consequently  en- 
deavor to  avert  this  possible  though  erroneous  impression. 


CONGENEKIC   MEKIDIANS. 


33 


In  the  convex  sphero-cylindrical  lens  of  double  form,  Fig.  25,  if  we 
consider  the  refraction  for  each  surface  independently  of  the  other,  we 
should  find  a  focal  point  at  F-  for  the  convex  spherical  surface,  s-,  and  a 
focal  line,  say  at  IF  I,  for  the  cylindrical  surface  c^.  Their  combination 
"living  rise  to  augmented  refraction  in  the  vertical  plane,  however, 
occasions  a  displacement  of  the  focal  line  I  F  Ho  the  position  of  V  F^  l\ 


Fig.  26. 
Convex  Sphero-cylindrical  Lens  (+  ^^  O  ^ —  d  axis  90°) — Periscopic  Form. 

The  final  rays  from  the  outermost  points,  e°,  in  the  horizontal  plane 
being  direcied  to  the  focal  point  F-,  it  is  evident  that  focal  line  l^  F^  P 
must  become  subject  to  the  influence  of  the  spherical  refraction  in  this 
plane,  thereby  establishing  the  points  d^  and  d^,  and  restricting  the  magni- 
tude of  the  focal  line  to  d^  F^  d\ 

The  final  rays  from  the  outermost  points,  e^,  in  the  vertical  plane,  which 
in  absence  of  the  cylinder  would  have  been  directed  to  the  focal  point  F^, 
now  cross  each  other  at  F^.  Their  extremities  are  therefore  displaced  from 
F-  to  d-  and  c?-,  thus  resulting  in  the  destruction  of  the  focal  point  F^,  and 
establishing  a  limitation  of  the  ravs  to  a  created  focal  line  d-  F-  d^. 


§  30.  The  convex  sphero-cylindrical  lens  of  periscopic  form.  Fig.  26,  is 
constructed  by  combining  a  weaker  concave  cylinder,  cS  with  a  convex 
spherical  surface,  s""-,  the  axis  of  the  cylinder  here  being  placed  in  the 
rertical  instead  of  the  horizontal  plane  for  the  purpose  of  future  reference. 

In  this  case  we  ha^e  given  to  the  spherical  surface  s-  a  curvature  corre- 
sponding to  the  focal  point  F^,  and  to  the  cylindrical  surface  cS  a  curvature 
which,  acting  in  combiaation  with  its  associated  horizontal  meridian  of  the 


34  COMPOUND  LENSES. 

spherical  surface,  causes  the  rays  to  unite  at  the  focal  line  d"^  F^  d?.  The 
reasons  given  for  the  destruction  of  the  focal  point  ¥^,  in  the  lens  Fig.  25, 
may  in  this  instance  be  similarly  applied  to  explain  the  creation  of  the 
primary  focal  line  d}  F^  d},  as  well  as  the  limitation  of  the  secondary  focal 
line  to  the  magnitude  d-  F^  d~. 


§  31.  A  characteristic  difference  between  the  double  and  the  periscopic 
form  of  astigmatic  lens  consists  in  the  fact  that  the  positions  of  the  focal 
lines  are  interchanged  with  respect  to  their  correlative  elements  of  creation. 
Thus,  in  Fig.  25  the  focal  line  d~  F^  d^  corresponds  to  the  initial  effect  of 
the  spherical  surface;  whereas,  in  Fig.  26  the  primary  focal  line  d^  F^  d^ 
corresponds  to  the  same. 

§  32.  This  difference,  however,  is  not  material,  as  it  is  evident  that  the 
magnitude  of  the  focal  lines  and  their  distance  from  the  lens  are  dependent 
upon  the  refraction  ascribed  to  its  two  principal  sections;  and,  since  any 
two  given  points  (d^  and  F-,  F'^  and  d^,  m}  and  nr)  definitely  fix  the 
position  of  a  line  or  ray  in  space,  it  is  further  obvious  that  the  direction  of 
all  final  rays  will  be  identical  for  any  lens*  in  which  the  right-angled 
coordinate  meridians  of  greatest  and  least  refraction  are  allotted  the  same. 

§  33.  To  demonstrate  the  analysis  of  formulae  for  these  equivalents, 
we  shall,  in  the  respective  figures,  designate  the  refraction  as  being  ex- 
pressed by 


la. 

4-  3.5  cyl.  axis  180° 

3  +  1.5  cyl.  axis    90°. 

(Fig.  24.) 

Ila. 

-|-  1.5  spherical 

3  4-     2  cyl.  axis  180°. 

(Fig.  25.) 

Ilia. 

+  3.5  spherical 

C—     2  cyl.  axis    90°. 

(Fig.  26.) 

It  being  necessary  to  become  thoroughly  familiar  with  the  meridians  of 
greatest  and  least  refraction,  it  is  considered  expedient  to  picture  these  in 
their  respective  planes  of  activity,  V  and  H,  as  shown  in  their  correlative 

*  Wherein  the  rays  are  incident  in  the  immediate  vicinity  of  the  optical  axis,  and  the  thick- 
ness of  the  lens  is  a  vanishing  quantity  in  proporiion  to  the  focal  distances  of  the  surfaces. 


CONGENERIC   MERIDIANS. 


35 


fiectional  diagrams,  Fig.  24a,  Fig.  25a,  Fig.  26a,  and  to  refer  to  them,  as 
follows : 


Fig.  24a.  Fig.  25a.  Fig.  26a. 

Formula  la.     +  3.5  eyl.  axis  180°  C  +  1.5  eyl.  axis  90°. 

Eefraction  =  1  _j_  3  5  vertical  C  +  1-5  horizontal  =  +  3.5V  C  +  1.5H. 
Fig.  24a.   )  1-^1 

Formula  Ila.     +  1.5  spherical  C  +  ^  cjl.  axis  180°. 

Eefraction :  |  _j_  g  _|_  ^  5  yertical  C  +  1-5  horizontal  =  +  3.5V  C  +  1.5H. 
Fig.  25a.    )  1^1 

Formula  Ilia.     +  3.5  spherical  3  —  2  eyl.  axis  90°. 

Eefraction  ^  1    1   3.5  vertical  3  —  2  +  3.5  horizontal  =  +  3.5V  C  +  1.5H. 
Fig.  26a.    i  ^  ^  -r  -r  —  t 

Pursuant  to  §  32  we  find  that  the  lenses  la,  Ila,  and  Ilia  are  asymmetri- 
cally-refracting equivalents. 

§  34.  As  the  preference  is  generally  given  to  the  double  form  (Formula 
Ila),  and,  under  certain  circumstances,  occasionally  to  the  periscopic 
(Formula  Ilia),  we  here  only  give  the  rules  applicable  for  the  conversion 
of  the  one  into  the  other  formula. 


To  convert  the  double  into  tlie  periscopic  form : 

Rule  1.  Place  the  sum  of  both  numerals  of  refraction  as  the 
numeral  for  the  newly-created  spherical  elements,*  and  com- 
bine with  the  same  cylindrical  element  having  its  sign  and 
axis  reversed. 


*The  sign  of  the  original  spherical  remaining  unchanged. 


36  COMrOUND  TvENSES. 

To  convert  the  periscopic  into  the  double  form : 

Rule  2.  Place  the  difference  of  both,  numerals  of  refraction  as 
the  numeral  for  the  newly-created  spherical  element,*  and 
combine  with  the  same  cylindrical  element  having  its  sign 
and  axis  reversed. 

§  35.  As  these  lenses  are,  practically,  only  nsed  for  the  correction  of 
anomalies  of  ocular  refraction,  it  is  customary  when  adapting  them  to  note 
tbe  positions  of  the  cylindrical  axes,  which  are  precisely  indicated  by  the 
graduations  of  the  trial-frame.  This,  however,  does  not  change  the  inherent 
properties  of  the  lenses,  whose  meridians  of  greatest  and  least  refraction 
are  always  90°  apart  for  all  possible  axial  positions  between  0°  and  180°. 

Thus,  in  the  instance  of  the  formula : 

+  1.5  sph.  C  -\-  0.50  cyl.  axis  130" 

the  periscopic  form  would  be  expressed  according  to  Kule  1,  §  34,  by 

+  2  sph.  C  —  0.50  cyl.  axis  40°. 

Inversely,  the  former  may  be  made  the  result  of  the  latter  by  application 
of  Rule  2,  §  34. 

A  table  showing  the  available  combinations  ])y  crossed  convex  cylinders, 
from  0.25D.  to  3.5D.,  is  hereto  appended,  wherein,  according  to  §  24, 
crossed  convex  cylinders  of  identical  curvature  are  substituted  by  their 
spherical  equivalents. 

The  diagonal  column  of  spherical  lenses  divides  the  table  into  two  sets 
of  compound  lenses  which  are  duplicates  in  refraction,  though  differing  in 
the  positions  of  their  cylindrical  axes  by  90°. 

Thus  all  lenses  in  the  vertical  columns  beneath  the  spherical  are  correla- 
tive duplicates  of  the  lenses  in  the  horizontal  columns  to  the  rigJit  of  the 
same  spherical.  (A'  =  a'),  {A^~  =  a^),  (A^  =  a'),  (B^  =  b'),  (B'  =  6*), 
(B^  =  &■''),  etc. 


*The  sign  of  tbe  original  spherical  remaining  unchanged. 


COMPOUND  LENSES. 

2.     CONGENERIC  MERIDIANS  (CONCAVE). 

§  36.  The  preceding  general  principles  are  alike  applicable  to  the 
similarly*  planned  concave  compound  lenses  Figs.  27,  28,  29,  in  eacli  of 
which  the  focal  lines,  and  consequently  also  the  focal  interval  and  circle  of 
least  confusion  are  virtual,  and  in  the  negative  region  before  the  lens. 

All  parallel  rays  incident  upon  and  within  the  periphery  of  the  circle  C, 
in  any  of  the  figures,  will,  therefore,  result  in  final  rays  behind  the  lens 
which  appear  to  emanate  from  correlatively  established  virtual  points  d^ 
and  F^,  F^  and  d-,  m^  and  m~  of  and  within  the  limits  of  the  focal  lines 
before  the  lens.  For  these  lenses,  respectively,  we  allot  the  refraction  as 
follows : 

lb.         —  1.5  cyl.  axis  180°    C  —  3.5  eyl.  axis  90°.     (Fig.  27.) 
lib.       —  1.5  spherical  C  —  3  cyl.  axis    90°.      (Fig.  28.) 
Illb.     —  3.5  spherical  Z-  +  2  cyl.  axis  180°.      (Fig.  29.) 

and  which,  by  a  similar  method  of  analysis  to  §  33,  pursuant  to  §  32,  will 
be  found  to  be  asymmetrically-refracting  equivalents. 

According  to  Kule  1,  §  34,  as  an  instance,  the  concave  sphero-cylin- 
drical lens 

—  1.25  sph.  ~  —  0.75  cyl.  axis  160° 

may  b?  converted  into  the  pcriscopic  form 

—  2  sph.  C  +  0.75  cyl.  axis  70°, 

and  vice  versa,  according  to  Ilule  2,  §  34. 


*The  meridian  of  greatest  refraction  is  here  placed  in  the  horizontal  instead  of  the  verticat 
plane. 

37 


38 


COMPOUND  LENSES. 


mV^' 


a> 


Fig.  27. 
Concave  Cylindro-cylindrical  Lens  ( —  c*  axis  180° 


c^  axis  90°). 


d> 


d2 


^,Ar:^^ 

«• 

el   \      \ 

.L_.  e" 

ft 

'Z^>^^i 

i 

!      .G 

Fig.  28. 
Concave  Sphero-cylindrical  Lens  {—  s^  ^  —  d^  axis  90°) — Double  Form. 


Fig.  29. 
Ooncavs  Sphero-cylindrical  Lens  ( —  s^  O  +  c^  axis  180°)  —  Periacopic  Form. 


In  the  above  figures,  i",  e",  and/"  are  associated  with  horizontal,  and  i»,  c*,  and/* 

with  vertical  refraction. 


COMPOUND  LENSES. 

3.    CONTRA-GBNERIC   MERIDIANS  (CONVEX  AND   CONCAVE). 

§  37.  Hitherto  we  have  considered  different  amounts  of  refraction,  re- 
stricted to  the  same  type,  convex  or  concave,  for  the  principal  right-angled 


Fig.  30. 
Concavo-convex  Cylindro-cylindrical  Lens  { —  c'  axis  90°  ':i^  -[-  c^  axis  180°). 


sections.  In  contradistinction  thereto,  and  as  a  final  complication,  we 
may  combine  in  a  lens  different  or  even  like  degrees  of  refraction,  though 
of  opposite  type;  namely,  convex  in  one  and  concave  in  the  other  diamet- 
rically-opposed coordinate  meridian.  As  an  instance,  we  may  select  the 
compound  lens  Fig.  30,  represented  as  consisting  of  a  plano-concave,  c% 
and  a  plano-convex  cylinder,  c",  so  combined  as  to  place  their  active  me- 
ridians at  right  angles  to  each  other. 

39 


40 


COMPOUND  LENSES. 


Independently  considered,  each  cylinder  c^  and  c-  would  have  its  focal 
line  Z^  F^  l^,  and  V^  F-  P,  of  original  magnitude  in  the  region  of  its  sign  — 
and  +  respectively,  and  consequently  on  opposite  sides  of  the  lens. 


k^- 


Fig.  30. 
Concavo-convex  Cylindro-cylindrical  Lens  ( —  c'  axis  90°  C:  +  c^  axis  180°). 


When  the  cylinders  are  associated,  however,  the  final  rays,  which  would 
have  been  restricted  to  the  limits  of  the  focal  line  F  F~  P  for  the  cylinder 
c^  will,  by  virtue  of  the  dispersive  effect  of  the  cylinder  d  in  the  horizontal 
plane,  be  confined  to  an  augmented  focal  line  d-  F~  d~,  within  the  limits 
d--d~,  for  the  outermost  rays  emanating  from  the  point  F^  of  the  virtual 
focal  line,  P  F'  I\ 

By  a  similar  method  of  reasoning  to  §  2G,  all  final  rays  within  the  limits 
of  the  circle  C  will  be  accorded  associated  vertical  and  horizontal  refraction 
culminating  in  their  united  intersection  of  a  line  d~  F~  d-,  of  the  hori- 
zontal plane  in  the  positive  region  behind  the  lens.  Interception  of  these 
rays,  by  successive  transverse  vertical  planes,  will  make  manifest  a  dem- 
onstration of  similarly  arranged  ellipses,  respecting  their  greatest  and  least 
diameters,  before  and  behind  the  focal  line  d"^  F-  d-.  By  projecting  the 
final  rays  into  the  region  of  their  apparent  emanation  from  before  the  lens, 
we  would  obtain  a  similar  increase  of  the  virtual  focal  line  V-  F^  V-  to  the 
magnitude  d^  F^  d^,  and  to  a  reversal  of  the  so-defined  ellipses,  respecting 
their  greatest  and  least  diameters,  as  shown  by  the  dotted  lines  in  the  nega- 
tive region  (Fig.  30). 


CONTKA-GENEKIC    MEK1DIAN8. 


41 


§  38.  Identical  refraction  is  also  preferably  obtained  in  this  instance  bj 
eombinationp  of  spherical  and  cylindrical  surfaces. 


Fig.  31. 
Concavo-convex  Sphero-cylindrical  Lena  (+  s^ 


r'  axis  90°) 


The  combination  of  a  convex  spherical  surface  with  the  active  meridian 
of  a  stronger  concave  cylinder  creates  a  periscopic  section  which  is  concave ; 
whereas  the  combination  of  a  concave  spherical  surface  with  the  active 
meridian  of  a  stronger  convex  cylinder  results  in  a  periscopic  section  which 
is  convex.  The  identity  of  the  refraction  for  these  combinations  becomes 
apparent  by  reference  to  the  concavo-convex  sphero-cylindrical  lenses  Figs. 
31  and  32,  in  which,  by  a  judicious  selection  of  the  respective  spherical 


ipi 


..i--""    « 


U-'- 


Fis.  32. 


Concavo-convex  Sphero-cylindrical  Lens  ( — s^  O  +  ^*  axis  180°). 

and  cylindrical  curvatures,  according  to  §  32,  the  demanded  positive  and 
negative  elements  of  refraction  for  the  principal  meridians  of  the  crossed 
cylindrical  lens,  Fig.  30,  are  fulfilled. 


42 


COMPOUND  LENSES. 


To  illustrate  the  equality  of  formulae  characterizing  these  equivalents, 
we  refer  to  their  correlative  sectional  diagrams  Figs.  30c,  31c,  32c,  in  the 
order  following : 


Fig.  30c. 


Fig.  31c. 


Fig.  32c. 


Formula  Ic.  —  1.5  cyl.  axis  90°  C  +  3.5  cyl.  axis  180°.  (Fig.  30.) 

Refraction :  )  ,      .  .    _^ 

Fio'    30c      r  —        horizontal  .^  -\-  3.5  vertical  =  —  1.5H  3  +  3.5V. 

Formula  lie     -f  3.5  spherical  C  —  5  cyl.  axis  90°.     (Fig.  31.) 
FirtlT  *  1  —  5  +  3.5  horizontal  C  +  3.5  vertical  =  —  1.5H  C  +  3.5V. 

Formula  IIIc.     —  1.5  spherical  C  +  5  cyl.  axis  180°.     (Fig.  32.) 

Fig.  32c.     \ 
Eefraction :  j 


1.5  horizontal 


1.5  +  5  vertical  =  —  1.5H  C  +  3.5V. 


§39.  These  lenses  being  equivalents  (see  §  32),  we  here  give  the  only 
rule  required,  for  reasons  later  given  (§40),  for  converting  the  cylindro- 
cylindrical  lens  (Formula  Ic)  into  the  concavo-convex  sphero-cyHndrical 
lenses  (Formulae  lie  and  IIIc). 


Rule  3.  Place  the  sum  of  both  numerals  of  refraction  as  the 
numeral  of  the  newly-created  cylindrical  element,  giving  to  it 
both  the  sign  and  axis  of  either  cylinder,  and  combine  with  the 
neglected  cylindrical  numeral  and  its  associated  sig^  as 
spherical. 


CONTKA-GENEKIC    MEKIDIAIS'S.  43 

Comparison  of  the  iieriseopic  lenses  Figs.  26  and  29  with  the  lenses  Figs, 
31  and  32,  respectively,  exhibits  a  striking  similarity  in  construction.  The 
characteristic  difference  between  them  is  that  in  the  latter  the  cylindrical 
exceeds  the  spherical  refraction,  whereas  in  the  former  the  reverse  is  the 
case. 


§  40.  In  a  case  of  mixed  (contra-generic)  astigmatism,  demanding  the 
foregoing  correction,  it  becomes  necessary  to  determine  the  chief  meridians 
—  1.5  and  -\-  3.5  independently  of  each  other,  thereby  obtaining  the  com- 
bination expressed  by  Formula  Ic,  as  by  an  endeavor  to  correct  through 
introducing  a  spherical  element  in  any  proportion  or  wholly  of  either 
equivalent  (Formula  lie  or  IIIc),  an  improvement  in  one  meridian  would 
always  be  attended  by  a  proportionate  derangement  in  the  other,  with  a 
probability^  of  the  patient  failing  to  appreciate  the  benefits  of  its  applica- 
tion. 

It  is  only  in  consequence  of  this  fact  that  the  lenses  of  the  Formulae  lie 
and  IIIc  are  rarely  the  direct  result  in  subjective  optometry,  whereas,  in 
cases  of  regular  astigmatism  with  congeneric  meridians,  the  lenses  Ila,  Illa 
and  lib,  1 1  lb  arc  most  apt  to  be. 

§  41.  Astigmatism  has,  in  the  main,  been  attributed  to  asymmetry  of 
the  cornea,  though  the  crystalline  lens  is  often  found  to  be  implicated;  yet, 
specifically,  in  a  case  of  mixed  astigmatism,  in  which  the  crystalline  lens 
does  not  assist,  it  is  improbable  that  the  corneal  surface  can  ever  be  of  the 
form  requisite  to  include  reversed  curvatures.  Fig.  3G.  In  such  instance 
the  ametropia  is  rather  more  apt  to  be  one  in  which  an  opposite  type  of 
astigmatism  is  in  excess  of  an  existing  hypermetropia  or  myopia,  respect- 
ively. Accepting  tliis  to  be  the  case,  such  an  eye  would  fall  heir  to  the 
features  accredited  to  hypermetropia  or  myopia  respecting  the  "nodal 
points"  and  "amplitude  of  accommodation;"  wherefore,  in  prescribing 
either  of  the  aforesaid  sphero-cylindrical  equivalents,  a  preference  might  be 
given  to  that  form  wliich  would  be  commensurate  with  the  inherent  physical 
and  physiological  developments  above  alluded  to. 


TORIC  LENSES. 

CONGENERIC  AND  CONTRA-GENERIC  MERIDIANS. 

§  42.  The  properties  of  astigmatic  refraction  are  also  fulfilled  in  a  lens 
by  creating  for  it,  opposite  to  its  plane  side,  a  single  surface  whose  diamet- 
rically opposed  principal  meridians  are  of  unequal  refraction. 


Fig.  33. 


Fig.  34. 


Such  a  surface,  called  a  torus,  is  shown  in  Fig,  33,  wherein  the  curva- 
ture c^  of  the  radius  r^  and  refraction  3D.  is  rotated  upon  a  vertical  axis  R 
so  as  to  create  the  curvature  c^  whose  radius  r-  is  chosen  to  produce  2D. 

In  Fig.  34  two  lenses  are  shown  to  be  included  w^ithin  the  surface  so  de- 
veloped and  an  opposite  plane  side,  the  one  being  a  plano-convex  toric 
lens  L^ — the  other  a  i)lano-concave  toric  lens  L-. 

From  the  construction  it  follows  that  these  lenses  are  each  possessed  of 

3D.  of  refraction  in  the  vertical,  and  2D.  of  refraction  in  the  horizontal 

meridian,  so  that  the  formulae  for  the  same  may  be  expressed  by 

44 


TOEIC     LE^'SES.  45- 

(AJ      [+  r>D.  Ref.  90°  Z  +  3D.  Ref.  180°]  Tor {L') 

(Bj)      [— HD.  Rof.  90° '2  — 2D.  Ref.  1S0°]   Tor (/>=) 

as  a  distinction  to  the  correlative  fomuila}  Ay  and  B^  for  a  pair  of  crossed 
cylinders  of  identical  refraction, 

(A.)      +  ;5  cyl.  axis  180°  C  +  2  cyl.  axis  90° 
(Ba)     —  3  cyl.  axis  180°  C  —  2  cyl.  axis  90° 

and  their  sphero-cylindrical  equivalents,  respectively : 

(  +  2  sph.  C  +  1  cyl.  axis  180°    (Double  Form). 
^    '  1  +  3  sph.  C  —  1  cyl.  axis    90°    (Periscopic  Form). 

\  —  2  sph.  C  —  1  cyl.  axis  180°    (Double  Form) . 
^    ''^  i  —  3  sph.  C  +  1  cyl.  axis    90°    (Periscopic  Form). 

§  43.  The  rotary  body  shown  in  Fig.  33  may  also  be  considered  to  have 
"been  created  by  bending  a  simple  cylindrical  lens  c^  to  the  radius  r-. 

In  such  an  attempt,  the  lens  of  the  Formula  A^  might  1)e  obtained  by 
bending  a  3D.  cylindrical  lens  to  that  radius,  which  effects  a  refraction  of 
2D.,  or  a  2D.  cylindrical  lens  to  a  radius  producing  a  refraction  of  3D. 
In  the  latter  case  the  lens  would  merely  require  to  be  turned  90°  so  as  to 
correspond  with  the  rest  of  the  formulae  The  inner  or  back  surface  would 
naturally  also  require  to  be  restored  to  a  plane,  as  indicated  by  the  dotted 
parallelogram  in  Fig.  33. 

The  suggested  method  being  impracticable,  the  process  of  grinding  must 
be  resorted  to;  although  this  at  present  involves  more  complicated  ap- 
paratus. A  lens  having  one  surface  spherical  and  the  other  toric  is  called 
■&  sphero-toric  lens.     Its  advantages  are  explained  in  a  subsequent  paper. 


§  44.  A  toric  surface  with  contra-generic  meridians  is  shown  in  Fig.  35, 
wherein  the  concave  section  c^,  of  the  refraction  —  3D.,  is  rotated  upon 
the  vertical  axis  R,  so  as  to  create  the  convex  curvature  c-,  whose  refrac- 
tion is  +  2D. 


46 


TOEIC     LENSES. 


In  Fig.  36  two  plano-toric  lenses  L^  and  L~  are  shown  to  have  the  same 
toric  surface  as  in  Fig.  35,  As  the  principal  meridians  in  each  of  these 
lenses  are  convex  and  concave,  we  may  write  their  formulas  as  follows: 

(Ci)      [—  3D.  Eel  90°  C  +  2D.  Kef.  180°]  Tor {U) 

(Di)      [+  3D.  Kef.  90°  C  —  3D.  Kef.  180°]  Tor (L=) 

so  as  to  distinguish  them  from  the  correlative  formulae  for  crossed  cylin- 
ders of  identical  refraction : 

(Co)     —  3  cyl.  axis  180°  C  +  2  cyl.  axis  90° 

(Do)     4-  3  cyl.  axis  180°  C  —  2  cyl.  axis  90° 

and  their  sphero-cylindrical  equivalents,  which  are  respectively: 

j  +  2  sph.  C  —  5  eyl.  axis  180°. 
(^a)  I  _  3  gph.  C  4-  5  cyl.  axis    90°. 


j  —  2  sph.  C  +  5  cyl.  axis  180°. 
(^3)  I  -I- 3  sph.  C  —  5  cyl.  axis    90°. 

The  student  should  practise  transforming  optionally  chosen  formulae 
by  applying  the  rules  given  in  §  34  and  §  39,  when  he  may  refer  to  the 
appended  tables  to  verify  the  correctness  of  his  own  work. 


NUMERALS  OF 
Metric  System 

REFRACTION 
Inch  System 

Focal  Distances 

Focal  Distances 

Centimeters 

DiOPTRIES 

Approximates 

U.S. 
Standard  Inches 

400. 

0.25 

1:160 

157.V 

200. 

0.50 

1:80 

78| 

133.3 

0.75 

1:53 

52i 

100. 

I. 

1:40 

39| 

80. 

1.25 

1:32 

31 1 

66.7 

1.50 

1:26 

264- 

57.1 

1.75 

1:22 

22JI 

50. 

2. 

1:20 

ISri 

44.4 

2.25 

I:I8 

17.- 

40. 

2.50 

1:16 

15i 

36.4 

2.75 

1:14 

14t\ 

33.3 

3. 

1:13 

13i 

30.8 

3.25 

1:12 

12| 

28.6 

3.50 

I:II 

4 

25. 

4. 

1:10 

23.2 

4.50 

1:9 

4 

20. 

5. 

1:8 

7| 

18.2 

5.50 

1:7 

n 

16.7 

6. 

l:6i 

6A 

15.4 

6.50 

1:6 

6 

14.3 

7. 

l:5| 

54 

12.5 

8. 

1:5 

4B 

11.1 

9. 

1:41 

4| 

10. 

10. 

1:4 

311 

9.1 

II. 

1:31 

3^^ 

8.3 

12. 

l:3i 

3A 

7.7 

13. 

1:3 

3^^^ 

7.1 

14. 

I:2| 

8|| 

6-7 

15. 

2f 

6.3 

16. 

l:2| 

2iV 

5.5 

18. 

l:2i 

2t\ 

5. 

20. 

1:2 

IH 

2.5 

40. 

1:1 

If 

The  above  table  has  been  arranged  for  comparison  of  the  metric  with  the  old  system  of 
numbering,  in  which  1  inch  was  adopted  as  the  unit.  A  lens  of  10,  20  or  40  inches  focus 
is  therefore  represented  as  being  ■^^,  -^j^,  or  ^^  of  the  refraction  of  the  old  standard. 

The  focal  distances  have  been  calculated  upon  the  basis  that  1  meter  =^  100  centimeters  = 
39.87  U.  S.  standard  inches,  through  dividing  each  of  these  equivalents  by  the  dioptral 
numerals.  To  render  a  harmony  of  the  numerals  of  the  two  systems  possible,  it  is  found 
necessary  to  neglect  slight  fractional  variations,  as  shown  in  the  differences  between  the 
divisors  in  the  3rd  with  the  figures  of  the  4th  column.  1  dioptry  being  placed  as  equivalent 
to  5'^,  lenses  of  2,  3,  or  4  dioptries  may  be  calculated  as  ^*g  =  ^,  ^^  --^  -^j,  or  ^*g  =^  ^g,  respec- 
tively, without  m.iterially  conflicting  with  the  practical  demands  upon  accuracy  in  a  sub- 
stitution of  one  system  of  numerals  for  the  other. 

47 


DIOPTRIES 


+025C.90' 


-H).50C90 


+0.75C.90' 


+  l.00t.9  0" 


+  I.85CBTD' 


+  l.5Da90' 


+  f.75C90' 


-f£.OOC.9  0' 


+2.25C.90* 


+2.50C.9  0 


+27  5(190'' 


+3.00  ceo' 


+3.2  5C.0O° 


+3.500.90" 


In  the  a 
del 

With  the 
cy 


I.    TABLE  OF  CROSSED  CYLINDERS  AND  THEIR  SPHERO-CYLINDRICAL  EQUIVALENTS 
CONGENERIC  MERIDIANS  (CONVEX) 


OlOPTRIES 

+0.250  180° 

+0.500.160° 

+O75C.I80° 

+  1.00C.I  BO* 

+  I.25C.I  BO" 

+  IB0C.ia0° 

+  1.760.180" 

+2000180° 

+225tl60° 

+2S0C.1  80° 

+275C.1B0' 

+3.000.180° 

+325C.180° 

+3.S0C.IB0° 

+0250,90" 

B 

*'+0.2  50+02  5e 
+0.500-0250) 

*+0250+o.5  0e 

+0  7  30-0.5001 

'Va25O+07  5e 
+  I.0DO-075O) 

+0.2  50+1.006 
+  1  250-1  OOO) 

+025O+I256 

+  1.5  DO- 1250 

+0250+1.606 
+  1750-1. 500 

+0250+1756 
+2.OOO-1750 

+02  50+2.006 
+225O-aOO0 

+02  50+2256 
+25  00-2250 

+0250+2.506 
+2750-2500 

+0250+27  56 
+3.OOO-2750 

+0250+3,006 
+3250-3,000 

+02 50+32 56 
+3500-3250 

+<150C90° 

^+0.253+0  250) 
+050C-0.2  5e 

MM     ■ 

"+0.600+0256 
+0750-0250) 

"+0500+0506 
+  1  000-0500) 

"+0500+0756 
+  I25O-075O) 

+0.500+1.006 
+  1.5DO-I.OO0 

+0500+1256 
+  17  50-1.250 

+050O+I.506 
+2.OOO-I.5O0 

+  05  004-17  56 

+2250-1750 

+0S0O+2.006 
+2SOO-2iJO0 

+0500+2256 
+2750-2250 

+0500+2506 
+3000-2500 

+OSOO+2.750 
+1250-2750 

+05  00+3.006 
+3  500-3000 

+075C9D° 

*'+0.2  52+0.5  OH 
^075C-050e 

'+05  00+0.2  50) 
+0750-02  56 

0 

+07  50+02  56 
+  1.00O-0.2  5O) 

+0750+0506 
+  1.250-0.500 

+07  50+0756 
+  1,500-0750 

+  0750+1.006 
+  I7  5O-I.OO0 

+0750+1.2  56 
+2.ODO-1.250 

+  07  50+1506 
+  2250-1.500 

+07 50+ 17  56 
+25 DO- 17  50 

+C750+2.009 
+27  5O-2.OO0 

+0760+2256 
+3000-2250 

+0750+2506 
+3250-2500 

+07  50+2756 
+350O-275(I> 

+  I,00t9  0* 

*'+02  5C+075<D 

+  i.oo;-07  5e 

'+0.5  00+05  00) 
+1000-0506 

+  0750+0250) 
+  1.000-0256 

0 

+  1.000+0.256 
+  1.250-0^50 

+  1.000+0.506 
+  1.5OO-O.5O0 

+  1.0  00+0756 
+  1.7  50-0  7  50 

+  1,000+1.006 

+aooo-ioo0 

+  1.0  00+1.250 
+  2^50-1.250 

+  1.000+1.506 
+  2500-1.500 

+  1.000+1  756 
+2750-1750 

+  1  000+2006 
+3.OOO-2OO0 

+  1,000+2256 
+3250-2  250 

+1 .000+2506 
+35OO-2SO0 

+  I.E!CStl" 

+0.2  5;+I.OO(I) 
+  l.25C-l.00e 

'+0  5  00+07  50) 
+1250-07  56 

+075O+0.50O) 
+1250-0509 

+  I.00O+025O 
+1 250-0256 

e 

+  l.E5O+0.25e 
+  ISOO-O.250 

+  1.2  5O+0.506 
+  17  50-0  5  00 

+  1,250+0756 
+£000-0750 

+  1,250+1.006 
+EE5O-I.OD0 

+  1250+1.256 
+2SOO-I.250 

+  1.250+I.S06 
+27SO-I.5O0 

+1250+1756 
+300O-I 750 

+  12SO+2-006 
+325O-2J)O0 

+  1250+2256 
+3.5  00-2250 

+  1  50I190' 

+D£5;+I 250 
+  1503-1.2  56 

+  Q5  0O+I00O1 
+1500-1006 

+  07  50+07  50) 
+1 600-0756 

+1 000+0500 
+  1.5  00-0  606 

+  I.2  5O+O.250 
+  I.50O-0256 

(■') 

+  I.50O+0256 
+1750-0250 

+  1.600+0506 
+iODO-a5O0 

+  1.50O+07  56 
+2250-07  60 

+  1500+1.006 
+25OO-1.OO0 

+  1.500+1256 
+2750-1250 

+  1500+1,509 
+3J)0O-l  500 

+1500+1756 
+3250-1750 

+  15  00+2.006 
+35OO-2J)O0 

+  f.75C90' 

+0.25C+I.50(I> 
+  l75C-l.50e 

+  05  00+ 12  50) 
+  1750-1256 

+D75O+I.0  0O1 
+  1.750-1.006 

+  1  000+0750) 
+  17  50-07  56 

+  I.25O+O.SO0 
+1750-0506 

+  I.6OO+O2S0 
+  I75O-0.256 

(,.) 

+  1. 750+0256 
+2,000-0250 

+1750+0506 
+22  50-0500 

+  1750+07  56 
+26  00-07  50 

+  1  750+l.DOe 
+2760-1.000 

+1750+1256 
+300O-I 250 

+175O+I506 
+325O-15O0 

+1750+1756 
+3500-1760 

■tE.OOC.OD" 

+0.250+175(1 
+2.000-17  56 

+  O.5OO+I.5O0 
+2000-1.509 

+07  50+ 1250) 
+2.000-1.256 

+  1.00O+I00O) 
+2000-1.006 

+I25O+O750 
+200O-D756 

+  1.SDO+D5O0 
+  2000-0.506 

+  I75O+O.250 
+21)00-0256 

0 

+2.00O+0.256 
+225O-Q2S0 

+2.0  00+ 05  06 
+25OO-O5D0 

+2000+07 5© 
+2750-0  7  50 

+2,0  00+1,006 
+3,000-1  000 

+2.000+1256 
+3250-1250 

+Z00O+  1.506 
+3.5  DO- 1500 

+2-25  C.BO* 

+  02  5O+200(D 
+2.2  50-2.006 

+  0500+1750) 
+2250-1756 

+075O+1  500) 
+2250-1.506 

+  1.0  00+ 1.250) 
+  22  50-1,256 

+  I25O+1.OO0 
+  2250-1  006 

+  1.5OO+O7  50 
+2250-07  56 

+  1.7  5O+O.5O0 
+  2250-0506 

+2.000+0^50 
+  2250-0.256 

B 

+225O+02S6 
+25OO-O2S0 

+2250+0506 
+2750-0  6  00 

+2250+0756 
+3  0  00-07  50 

+2250+1.006 
+3250-1000 

+2250+1.256 
+35  00-1250 

+2  50  0.9  0° 

+02  50+22  Sir 
+2.500-22  56 

+0S0O+200O) 
+2500-2006 

+0750+1750) 
+  2.50O-I756 

+  1.00O+150O) 
+2.5  00-1506 

+1250+1 250 
+  2500-1256 

+  ISDO+1.OO0 
+  2500-1.006 

+  1.75O+O750 
+  25  00-07  56 

+2000+0500 
+2500-0506 

+22  50+012  50 
+2500-0250 

B 

+250:+0256 
+2  750-0250 

+2500+0506 
+3000-0500 

+2500+0756 
+3250-07.50 

+25  00+1.006 
+J5OO-1.DO0 

+2750.90° 

+  02  50+2.5  00) 
+2  7  50-2  506 

+  0500+2250) 
+27  50-22  56 

+07  50+21)00) 
+2  750-2006 

+1OOO+I750 
+  2750-1.756 

+  1.250+1500 
+2760-1  506 

+  15OO+I.250 
+2750-1 256 

+  1.75O+I.OO0 
+2750-1006 

+200  0+075  0 
+2750-0756 

+Z2 5  0+05  00 
+2750-0506 

+25 00+02 50 
+275O-D2  50 

-B 

+2750+0256 
+3OOO-O250 

+2750+0500 
+32  50-0500 

+27  50+0756 
+1500-0750 

+5OOC.9  0' 

+  02  50+27501 
+3.0  00-27  56 

+0500+2500) 
+3000-2  506 

+  07  50+2.250) 
+3000-2256 

+  I00O+200O) 
+J00O-2.006 

+1250+1750 
+  3.00O-I756 

+1 500+1 500 
+  3,000-1506 

+  1750+1  250 
+  3000-1.256 

+2.000+1.000 
+300O-I.006 

+2250+0760 
+3.000-0  756 

+2500+0500 
+3000-0500 

+275O+D2S0 
+3D0O-0256 

,  H-iom 

+3000+0256 
+3250-0250 

+3.0  00+0506 
+3500-0500 

+3.E5CiO° 

+  0.25O+3.00OI 
+3i5O-3D0e 

+0  500+27  50) 
+3250-27  56 

+07  5O+2.50O) 
+3  2  50-26  06 

+  1.O0O+ 2.250) 
+32  50-2256 

+  I25O+2.OO0 
+3250-2006 

+1 50  0+17  50 
+3,250-1  7  56 

+  175O  +  1.5O0 
+  3.250-1506 

+2OOO+I.250 
+3.250-1256 

+  2250+1.000) 
+3250-1 006 

+2500+0750 
+3250-07  56 

+2750+0500 
+3  25O-D506 

+300O+O25O' 
+12  50-0  256 

B 

+3250+0256 
+3.5OO-O.2S0 

+3.50C.9ef 

+  0J  60+3.250) 
+15  00-3256 

+  0.500+3000) 
+3500-31)06 

+0750+2750) 
+J50O-2759 

+  I00O+250O. 
+3500-2506 

+  1.250+1250 
+3500-2.256 

+  l,5OO+aOO0 
+3.50O-2DD6 

+  1750+1.750 
+3.500-1  756 

+2.OOO+1.5O0 
+350O-I506 

+2250+1250 
+3iD0-  1256 

+2500+1000 
+J5OO-1.OO0 

+2750+0750 
+J5DO-0756 

+300O+060O1 
+3500-0506 

+325O+O250 
+35  00-02  56 

0 

In    the    above    formulae    the    first    numerals    apply    to    spherical,  and  the  second   to  cylindrical   refraction.      In    the    appended  signs,   the  upright  and  horizontal    diameters  (  |   and  — )  of    the  circles 

denote  the  axes  90°  and  180°,  respectively. 
With   the  exception    of    the    diagonal    column    of    spherical    equivalents,    each    field    contains    both    the   double    and    periscopic    form    of    convex    sphero-cylindrical    equivalent.      For    crossed    concave 

cylinders  it  is  merely  necessary  to  reverse  the  signs  +  and  —  wherever  they  occur. 


DIOPTRIES 

-0.25  C.9  0° 


-050C.90' 


-0.7  5  C.9  0' 


-i.oocao 


-I.25C.90* 


■1.50C.90° 


•l.75C.gQ* 


-2.00  C.9  0* 


-2.2509  0° 


-2  5  0  C.9  0° 


-2.7  6  C.9  0° 


-3.00  C.9  0* 


-32  5  C.9  0* 


-3.50C90° 


In   the    abo 

the  a 

Practical  eqi 

For  crossed 


DIOPTRIES 

-0.25C.90° 


-0.500.90° 


-0.75C.90° 


-i.oocao" 


-I.25C.90' 


1.5  0  0.90° 


-I.75C.90* 


-2.00C.90* 


-2.2509  0° 


-25  0  0.90° 


-2.7  5C.90* 


-3.00C.90* 


-32  5C.90* 




-3.50C90° 

In   the    abo 

the  a 

Practical  eqi 

For  crossed 


II.     TABLE  OF  CROSSED  CYLINDERS   AND  THEIR  SPHERO-CYLINDRICAL  EQUIVALENTS 
CONTRA-GENERIC  MERIDIANS  (CONVEX  AND  CONCAVE) 


HOPTRIES 

+025C.I80' 

+  O50C.(80- 

+07 5 CI  eo' 

+  1000.180° 

+  1  25a;  80" 

+  1  500180° 

4 1,75 CI  80° 

+2.00  01  60° 

+2260180° 

41500,180" 

+2750180° 

+300C.I  80° 

+1260,180° 

+J60ai80° 

-025M0* 

+a25C-OiO® 
-tt25.-+0.50e 

+a.B0c-a7M) 

-0.25C+076e 

+075C-I.00O) 

-o25:+i.ooe 

+  1  000-1.250) 
-0.250+1.259 

41  250-1  500) 
-0.250+1509 

+  I.50O-175® 
-0,250+1, 7Se 

+  1  750-2,00® 
-0250+2  0  09 

+2D0O-225® 
-0250+2269 

42250-250® 
-025042509 

42500-275® 
-0,25042769 

+2750-300® 
-0250+3009 

43000-325® 
-0250+3259 

+1260-350® 
-0250+3509 

+1500-375® 
-0250+37  5© 

-OSOC.SO" 

+  0.25C-0.75<D 
-0£0C+075e 

+0.50C-I.D0<D 

-a5o:+i.ooe 

+07  5C- 1.250) 
-0.5  00+1.259 

+  1.00C-I.50O) 
-0500+1.509 

+  1250-1.750) 
-050041769 

+  I.6OO-200® 
-0,6  00+2,009 

+1750-225® 
-0.60O+2259 

+2,000-250® 
-050O+25D9 

+£25C-275® 
-05  0042759 

+250  0-300® 
-0,500+3,009 

+2760-325® 
-06 0  0+32 59 

43000-350® 
-05004360© 

+3250-17  5® 
-0500+3759 

+3500-4,00® 
-0500+4000 

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+  1000-1,750) 
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41.000-3000) 
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41,500-400® 
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42500-500® 
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-2.75C.30- 

+  025C-M0a) 
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41,000-375® 
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41,500-425® 
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42  600-525® 
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+  0.25C-3.25(I) 
-3.000+3.259 

+0.50C-3.50(Il 

-3.oo:+35oe 

+0760-3750) 
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41.000-41)00) 
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41,600-450® 
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42500-5500 
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-3^5  too" 

+o.2B:-3,5Da) 

-3.25C+350e 

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-3.25C+375e 

+0750-41)00) 
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4I.OOO-4260) 
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41.600-4,75® 
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4225O-550® 
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+3  0  0O-6.2  5® 
-3250+G.259 

+3250-650® 
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+350O-a75® 
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-3.5DC.90" 

+  0.25C-375(B 
-3.50C+375e 

+0.500-4  0  00) 
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+0  7  50-4.250) 
-3  50  0+4.2  59 

41.000-4.600) 
-3.50C44609 

+  1.260-475® 
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+  1  500-5,00® 
-3,50045009 

+  1750-525® 
-3500+5258 

+2000-550® 
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42250-576® 
-15  00+57  59 

4250O-GJ)0® 
-160046009 

+2750-626® 
-3500462SO 

+  3.00O-650® 
-350O+B50O 

+3250-6,75® 
-1500+6769 

+1500-7.00® 
-1500+700© 

In  the    above  formulas  the  first  numerals  apply  to  spherical,   and  the  second    to  cylindrical  refraction.     In  the  appended    signs,    the  upright  and  horizontal  diameters  (  |   and   —  )  of  the  circles  denote 

the  axes  go°  and  180",  respectively. 
Practical  equivalents  can  only  be  produced  when  the  numerals  of  the  formula  coincide  with  those  included  in  the  adopted  dioptral  series,  page  47. 
For  crossed  cylinders  having  the  axes  of  the  concave  cylinder  at  180°  and  of  the  convex  cylinder  at  90°  it  is  merely  necessary  io  reverse  the  axes  throughout. 


SECTION  II 


DIOPTRIC  FORMULA 
FOR  COMBINED  CYLINDRICAL  LENSES 


APPLICABLE  FOR 


ALL  ANGULAR  DEVIATIONS  OF  THEIR  AXES 


WITH   SIX   ORIGINAL  DIAGRAMS   AND   ONE   HALF-TONE   PLATE 


SECTION  11 


DIOPTRIC  FORMUL/E 
FOR  COMBINED  CYLINDRICAL  LENSES 


APPLICABLE  FOR 


ALL  ANGULAR  DEVIATIONS  OF  THEIR  AXES 


WITH   SIX   ORIGINAL  DIAGRAMS   AND   ONE   HALF-TONE   PLATE 


DIOPTRIC    FORMUL/E 

FOR  COMBINED 

CONGENERIC  CYLINDERS 


I.     DIOPTRIC   FORMULA 

FOR  COMBINED 

CONGENERIC  CYLINDRICAL  LENSES. 


1.     RELATIVE  POSITIONS  OF  THE  PRIMARY  AND  SECONDARY 
PLANES    OF   REFRACTION. 

In  the  following  theorems,  a  prior  knowledge  of  the  established  mathe- 
matical deductions  applied  to  lenses  for  parallel  rays  incident  in  the 
immediate  vicinity  of  the  optical  axis,  and  in  which  the  lens-thicknesses 
are  considered  vanishing  quantities  in  proportion  to  the  focal  distances  is 
taken  for  granted.  The  formulae  here  advanced  are  therefore  dependent 
upon  those  which  have  not  been  carried  beyond  first  approximations. 
Practically,  in  almost  all  cases  that  occur,  the  thicknesses  of  the  combined 
lenses  are  very  small  quantities  compared  to  the  other  dimensions  involved, 
so  that  we  shall  consider  the  cylinders  to  be  so  thin  that  their  centers 
actually  coincide,  and  in  which  case  the  focal  distances  are  to  be  counted 
from  a  plane  perpendicular  to  the  optical  axis,  in  the  optical  center  of  the 
combined  lenses. 

In  Plate  I,  two  combined  convex  cylindrical  lenses  are  shown,  which, 
though  somewhat  at  variance  with  the  prescribed  conditions  of  thickness, 
will,  however,  better  serve  to  make  our  subject  clear. 

The  dotted  circle  shown  within  the  lenses,  with  its  center  at  the  optical 
center  o^  shall  represent  the  plane  above  alluded  to. 

The  passive  or  axial  planes  of  the  cylinders  are  shown  by  dotted  paral- 
lelograms at  A  and  a,  bisecting  each  other  under  the  angle  Aoa  =  ^  in  the 

optical  axis  at  o ;    and  their  active  planes  of  refraction  C  and  f,  which  are 

53 


54  DIOPTRIC   FORMULAE. 

of  necessity  at  right  angles  to  their  correlative  axial  planes,  similarly 
bisect  each  other  at  the  same  point.     Hence,  <^  Coc  =  <^  Aoa  =  y. 

The  compound  lens  thus  presented  consists  of  two  congeneric  cylin- 
drical elements,  each  of  which,  independently  considered,  will  have  its 
corresponding  focal  plane,  which,  for  convenience,  we  may  term  an 
elementary  focal  plane  of  the  combination.  Thus,  E^  and  E.^,  at  the  focal 
distances  f^  and  /.,,  are  the  elementary  focal  planes  for  the  cylinders  C  and 
c,  respectively.  The  cylinder  C  will  consequently  have  the  proj^erty  of 
deflecting  a  ray,  incident  at  D,  perpendicularly  from  Z>j,  in  the  plane  E^, 
to  the  point  Zj  of  the  axial  plane  A^  Z^ ;  whereas,  the  cylinder  c  will  have 
the  property  of  deflecting  a  ray  incident  at  the  same  point,  perpendicularly 
from  Z>2,  in  the  plane  E^^  to  the  point  K^,  of  the  axial  plane  a^  o^. 

The  greatest  amplitude  of  deflection  for  C  will  therefore  be  D^Z^  in  the 
plane  E^,  and  for  c  will  be  D^  V^  in  the  plane  E^.  It  is  further  manifest 
that  the  refracted  ray  D  V^  V^,  contributed  by  c  only,  in  attaining  its 
greatest  deflection  D.^  F^,  in  the  plane  E.^,  will  penetrate  the  plane  E^  at  V^, 
and  in  it  present  a  proportionate  deflection  Z>i  V^. 

D^  Zj  and  D^  V^,  being  amplitudes  of  deflection  reduced  to  the  same  plane 
^j,  will  then  bear  the  same  relation  to  each  other  as  their  corresponding 
refractions.      Thus 

or,  Z>iZ,  =  y-,    when  D^V^  =  -j-, 

J\  Jn 

and  which  may  easily  be  shown  to  be  the  case  when  the  deflections  are 
measured  in  a  plane  one  inch  from  the  lens.* 

Provided,  therefore,  that  the  deflections  are  measured,  within  the  same 
plane,  from  a  point  D^  of  the  same  line  of  incidence  DD^,  we  may  deter- 
mine the  resultant  of  two  deflections  D^Z^  and  D^V^^  for  any  angular 
deviation  existing  between  them  at  A,  by  the  physical  law  governing 
similarly  united  forces.  E>^M^,  as  the  diagonal  of  the  parallelogram 
A  Fj  M^  Z^,  will  consequently  be  the  resultant  deflection  accruing  from  a 
combination  of  the  cylinders  C  and  c. 

♦Refraction  and  Accommodation  of  the  Eye,  by  E.  Landolt,  M.D.,  Paris,  translated  by  C.  M. 
Culver,  M.A.,  M.D.,  Philadelphia,  1885  (see  page  58). 


CONGENERIC   CYLINDBttS.  55 

As  each  cylinder  contributes  a  plane  of  active  and  one  of  passive 
refraction,  we  shall  evidently  obtain  two  resultant  principal  planes  for  their 
combination,  the  one  of  greatest  refraction,  commonly  called  the  primary 
plane,  DD^o^o,  intersecting  the  angle  Coc  =  y  between  the  active  planes  of 
refraction  C  and  c,  and  one  of  least  refraction,  termed  the  secondary 
plane,  dd./).^o,  intersecting  the  angle  Aoa  =  y  between  the  passive  or  axial 
planes  A  and  a. 

The  primary  plane,  in  penetrating  the  plane  E^,  will  consequently  divide 
the  angle  C^o^c^  =  Coc  =  y  into  D^o^c^  =  «  and  Z>i^iC,  =  /5.  In  the  plane 
£^  we  shall  then  find  the  angles  a  and  /3  to  be  directly  dependent  upon  the 
associated  deflections  D^Z^  and  Z?,  V^  for  the  point  D^.  In  the  plane  £^  a 
similar  division  of  the  angle  A.^o,a.^,  by  the  secondary  plane,  will  be 
rendered  dependent  upon  d.,v.^  and  d.^z.^  for  the  point  d.^.  As  to  this,  the 
diagram  is  believed  to  be  sufficiently  clear,  without  further  reference. 

Since  the  resultants  D^Jlf^  and  d^?^i.,  define  the  directions  of  the  refracted 
rays  DAf^  and  dm.^,  it  is  further  evident  that  for  D  and  d  to  be  points  of 
the  primary  and  secondary  planes,  respectively,  they  will  have  to  be  so 
chosen  that  D^M^  and  d.^m^  shall  be  directed  towards  the  optical  axis  oo^o.,_ ; 
and,  as  we  shall  later  learn,  this  is  but  one  of  the  restrictions  which  renders 
a  diagram  somewhat  difficult  of  construction.  The  resultant  deflections 
Z>i  M^  and  d.pi.,  are  therefore  shown  in  the  primary  plane,  coincidetit  with 
Z?,^i,  and  in  the  secondary  plane  coincident  with  d.p.j^^  respectively. 

For  all  intermediate  points  of  the  circle,  the  resultant  deflections  deviate 
from  the  optical  axis.  This  has  been  taken  advantage  of  in  constructing 
Dr.  Burnett's  models,  and  in  determining  the  directions  of  twelve  refracted 
rays  in  each  of  the  figures  2,  Plates  II  and  IV. 

The  position  of  the  primary  plane  DD^o^o,  shown  as  dividing  the 

angle  C^o^c^  =  y  bo  that 

r  =  «  +  /5, (1) 

will  then  be  determined  by  fixing  the  relations  existing  between  «  and  /?. 

In  the  plane  E^,  from  the  triangle  D^Z^M^,  we  have 

D,Z,  :  Z,i7/,  =  sin  <  Z,M,D,  :  sin  <  Z,Z>,/7/,, 
<  Z,M,D,  =  <  D,o,c,  =  a, 


56  DIOPTRIC   FORMULAE. 

by  parallelism  of  Z^M^  and  c^o^ ;  and,  for  similar  reasons, 

.*.     D^Z^  :  Z^M^  =  sin  a  :  sin  /J, 

.-.     Z?iZi :  D,  Fj  =  sin  «  :  sin  z? (2) 

In  the  oblique  plane  DD^  V.^  we  find 

D,  V,  ■D,V.,=  DD,  :  DD,  ; 

ar,  as  DD^  and  Z?/?^  are  the  focal  distances  f^  and  f.^  of   the  cylinders  C 
aad  <r,  respectively, 

A^i:A^'.=/:/. (3) 

Multiplying  the  equations  (2)  and  (3),  we  obtain 

Z>jZj      sin  a  /i 


D,V^      BiiiiS  f^ 


W 


Since  D^o^  is  the  radius  of  the  circle  indicated,  we  may,  for  convenience, 
ascribe  to  it  the  value  1.     We  shall  then  have 

Z?jZj  =  sin  <^  A^i'2'i, 
<  A^i^i  =  C,o,Z,  —  <  n,o,C,. 
.  •.    <  n,o,Z,  =  90°  —  fi. 

.-.     D,Z^  =  sin  (90°  —  ;5)  =  cos  ,5.  ...     (5) 
In  the  plane  A  ^^  similarly  find 

AF,  =  sin  <^n,o,v„ 

.-.    <^D,o,V,  =  90°— «. 

.-.     Z?,F,  =  sin  (90°—  «)  =  cos  a.       .     .     (6) 


CONGENERIC    CYLINDERS.  57 

Substituting  the  values  for  Z?,  Z,  and  D^  V.^  from  (5)  and  (6)  in  the  equa- 
tion (4),  we  obtain, 

cos  /J      sin  a  yj 

cos  a      sin  /?  f., 
or,  by  multiplying  both  members  of  equation  by  2  and  transposing, 

2  cos  iS  sin  /?  =  2  cos  «  sin  «  ^^ 

.  •.  sin  2/3  =  sin  2a  4- (7) 

The  position  of  the  secondary  plane  dd^op^  shown  as  dividing  the 
angle  A^o.fl^  =  y  into  d^o^a.^  =  a  and  d^o^A^  =  ,5,  provided  d^o^  is  perpen- 
dicular to  D^o^,  will  be  determined  by  similarlj'^  fixing  the  relations  between 
a  and  /?.     Here  it  can  also  be  shown  that 

d^2   :  d^v.2  =  cos  «  :  cos  ,3 (8) 

2 

d,2^  :flf,0,  =/  :/, (9) 

d^z^  =  sin  /? (11) 

d.,v.^  =  sin  a (12) 

whereby,  as  before,  sin  2/9  =  sin  2a  ^  . 

We  therefore  conclude  that : 

1.  The  primary  and  secondary  planes  of  refraction  are  at  right 
angles  to  each  other  for  any  angular  deviation  of  the  axes  of 
two  combined  congeneric  cylindrical  lenses. 

In  a  further  consideration  of  the  relation  (7),  sin  2/3  =  sin  2  a-=y^,weob- 
serve  that  the  sines  of  the  angles  2a  and  2/3,  which  are  each  always  less  than 
90°,  merely  differ  by  the  co-efficient  ^--. 

If,  therefore,  /^  =  /^,  which  is  the  case  when  the  cylinders  are  of  equal 
refraction,  the  sin  2/3  will  be  equal  to  the  sin  2«,  and  which  can  only  be  the 
case  when  «  =  ^3,  or,  as  «  +  /3  =  r,  when  «  =  ,?  =  -L--  hence  ; 


58  DIOPTRIC    FORMULA. 

2.  For  combined  congeneric  cylinders  of  equal  refraction,  the 
primary  plane  equally  divides  the  angle  between  the  active 
planes  of  the  cylinders,  and  the  secondary  plane  similarly  di- 
vides the  angle  between  the  axial  planes  of  the  cylinders. 

In  case,  however,  /j  >  /,,  which  is  the  case  when  the  refraction  of  the 
cylinder  C  is  greater  than  r,  then  sin  2«  >  sin  2/3,  or,  when  «  >  /3,  so  that 

3.  For  combined  congeneric  cylinders  of  unequal  refraction, 
the  primary  plane,  in  dividing  the  angle  between  the  active 
planes  of  the  cylinders,  will  be  nearer  to  the  active  plane  of  the 
stronger  cylinder,  and  the  secondary  plane  consequently  nearer 
to  the  axial  plane  of  the  same  cylinder. 

This  is  also  demonstrated  in  the  diagram. 

As,  for  a  combination  of  two  cylinders,  C  and  r,  under  given  angular 
deviation  of  their  axes,  the  only  known  quantities  will  beyj,y^,  and  r^  it 
will  be  necessary  to  express  «  and  /5  in  terms  ofy],  /^,  and  y. 

This  is  accompHshed  through  the  equations  (1)  and  (7)  : 

r  =  «  +  /5 
sin  2/9  =  sin  2a  A^ 

when,  after  proper  substitution  and  reduction,  we  obtain  : 


/I    ,    1             /,  +/,  cos2r  ,,. 

cos  «  =  -v/  6  +  o"  — ^  .     .     .    (I) 


It  will  be  unnecessary  to  seek  ,9  in  the  same  manner,  since,  through  (1), 
we  find  ^  =  Y  —  "• 

When  reducing  this  formula,  for  any  given  value  of  ^,  pursuant  to  rea- 
sons later  given,  it  should  be  observed  thaty,  >  /,,  in  which  case  a,  within 
the  angle  y^  is  to  be  counted  from  the  axis  of  the  weaker  cyHnder. 


CONQENERIC    CYLINDERS.  59 


2.     POSITIONS   OF   THE    PRIMARY    AND   SECONDARY    FOCAL 

PLANES. 

As  the  plane  DD^o^o  is  the  primary  plane,  it  follows  that  all  parallel 
rays  incident  in  it  between  D  and  o  will,  after  refraction,  intersect  the  op- 
tical axis  ooj  at  some  point,  which  will  be  a  point  of  the  primary  focal  line. 
Therefore  the  resultant  ray  DM^M^,  in  attaining  its  greatest  deflection 
Z),v?/j  in  the  elementary  plane  E^,  will  establish  the  position  of  the  primary 
focal  line,  through  its  previous  intersection  of  the  optical  axis  oo„  at  the 
point  Oy 

In  the  secondary  plane  dd^o^o,  for  similar  reasons  O.^  will  be  a  point  of 
the  secondary  focal  line,  though  this  point  of  intersection  of  the  final  ray 
dm^m.^  with  the  optical  axis  is  more  distant,  in  consequence  of  the  inferior 
deflection  d^m^  in  the  plane  E,. 

Similar  resultant  deflections,  at  opposite  cardinal  points  of  the  circle 
within  the  lens,  define  the  directions  of  their  corresponding  refracted  rays. 
These  rays  not  only  limit  the  major  and  minor  axes  of  the  ellipses  shown 
in  the  planes  E^  and  E.^^  but  also  determine  the  lengths  of  the  focal  lines  at 
6>j  and  O.^.  Thus  O.^M^  represents  one-half  of  the  secondary  focal  line  at 
O^.  The  primary  focal  line,  in  the  secondary  plane,  perpendicular  to  YO^ 
at  0„  has  been  omitted,  to  avoid  possible  misinterpretation  of  more  impor- 
tant points  of  reference  in  the  diagram.  All  rays  parallel  to  the  optical 
axis,  incident  at  intermediate  points  of  the  circle  within  the  lens,  will, 
upon  refraction,  intersect  the  planes  E^  and  E.^  at  correlative  points  of  the 
ellipses  drawn  thereon. 

The  circle  of  least  confusion,  T,  will  lie  between  the  planes  E^  and  E^. 
(See  Plate  II,  Fig.  2. )  Its  position  may  be  determined  through  a  simple 
formula  given  by  Prof.  W.  Steadman  Aldis,  of  the  University  College, 
Auckland,  New  Zealand,  in  his  discussion  of  the  focal  interval  resulting 
from  rays  obliquely  incident  upon  a  spherical  lens.  * 

Our  object  being  to  determine  the  distances  of  the  primary  and  second- 
ary focal  lines,  or  planes,  from  the  principal  plane  within  the  combined 
cylinders,  we  shall  proceed  as  follows  : 

*Elementary  Treatise  on  Geometrical  Optics,  W.  S.  Aldif?,  M.A.,  Cambridge,  1886  (kcc  papeSO). 


€0  DIOPTRIC   FORMULA. 

In  the  primary  plane  DD^M^^  we  have 

DY:  DD,  =   YO,  :  D,M,. 
Substituting,  DY  =  O^o  =  F^     as  the  primary  focal  distance; 

YO^  —  D^o^  =  radius  =  1. 
.     F  =  -  -^^ (26) 

In  the  parallelogram  D^  V^M^Z^,  the  angle  between  the  forces,  D^  V^  and 
Z>iZj,  being  equal  to  <^  C^OyC^  =  y,  we  have,  as  the  resultant  deflection, 


D,M,  =  V  {D,z,f  +  (A  y,y  +  2  (A  K)  (Ai^i)  cos  r,        (27) 
in  conformity  with  the  statical  formula, 


R=  ^/  pt^  0^^  2.PQ  cos  r, 

for  forces  P  and  j2>  acting  at  the  same  point,  within  the  same  plane,  under 
the  angle  y. 

Substituting  in  (27)  the  value  of  D^Z^  =  cos  ii,  from  (5) ;  and  of  A  K 

f  f 

=  "—  A  ^r  from  (3),  =  —-  cos  a,  from  (6),  we  obtain, 

D^M^  =  -i/cOS^  /3  +  (f^)    COS^  a  +   2  -^y-  cos  a  COS  /S  COS  ^. 

Introducing  this  value  for  D^M^  in  (26), 

F,  =  ^'  •    ■     .     (28) 

I  //  \2  / 

-1/  COS^  /5  +    (  "y- )    COS'^  a  -\-  2  ~  cos  a  COS  /3  COS  7- 

By  substituting  the  proper  values  for  a  and  i5,  from  equation  (1),  after 
adequate  reduction  we  obtain 

17 J\Ji , 

^^\  [/.  C/i  +/.  COS 2r)  +  (/.  +/,)  (/,  +  >//M^2/y:  cos  2r  +/*)] 

(30) 


CONOBNERIC    CYLINDERS.  61 

Transforming,  and  substituting  1  —  2  sin^  r  for  cos  2)',  we  may,  for  con- 
venience in  calculating,  preferably  write 


F. 


/i/^  .       (II) 


^(/i±/Z-/j,^  sin=^  r  +  (/,  +/,)  ^l^-^^^'^-^-AA  sin'  r 

When  the  cylinders  are  of  equal  refraction,  y"^  being  equal  to/j  ==/,  the 
above  assumes  the  simple  form, 

F,=  —~^—~ (IV) 

1  +  cos  y 

In  the  secondary  plane  dd.,XO,,  we  have 
dX :  dd,  =  XO,  :  djn,. 
Substituting,  dX  =  0.,o  =  F,  as  the  secondary  focal  distance; 

dd,=/,; 
XO.^  =  radius  =  1. 

.-.    F,=  -f- (31) 

In  the  parallelogram  d^v^m^s.^,  the  angle  between  the  forces,  d^v^  and 
d^.,,  being  equal  to  <^  v./l.^z.^  =  180°  —  <^  A,o,a.,  =  180°  —  y, 


.  •  •    djn,  =  t/  {d,zj'  +  (d,v,y  +  2  id,v.,)  (d,z,)  cos  (180°  —r). 

Substituting  the  value  for  d.^z.^  =  -y-  d^z^,  from  (9),  =    ^  sin  /?,  from 
(11);  and  ior  d.^).,  =  sin  «,  from  (12),  we  obtain, 

d^m^  =  -*/  ("y-  )  sin-  /?  -f  sin^  a  —  2~  sin  a  sin  /3  cos  y] 

i^rhich,  introduced  in  (31)  and  being  multiplied  in  the  numerator  and  de- 
nominator by  ~ ,  gives 


62  DIOPTRIC    F0RMULi?5. 

/■ 

-* /sin'  ,3  +  ("t")   sin^  ''-  —  2  y   sin  'x  sin  ^  cos  /- 

Through  substitution  of  the  proper  vahies  for  «  and  /J,  from  equation 
(1),  after  suitable  reduction  we  find 

F^  /-/- ^ 

^lU.  (A  +A  cos  2j')  +  (/,  +/.)(/.~i//,^+2/,/,  cos  2r+//)] 

(33) 

Substituting,  cos  2/'  =  1  —  2  sin^ ;-, 

^/(A+A)^  -/,/,  sin^  r  -  (/,  +/.)^^-^'"±^'  -/,/.  sin'  , 

(HI) 

This  formula,  reduced  for  cylinders  of  equal  refraction,  /^  being  equal  to 
y^  =  /,  becomes 

F,  =  ^^ (V) 

^        1  —  cos  y 

It  may  be  of  interest  to  note  that  these  formulas  diSer  from  those  given 
for  F^  merely  by  the  minus  sign  in  the  denominator. 

The  preceding  formulae  being  alike  applicable  for  combinations  of  con- 
vex or  concave  cylinders,  the  fociyi  andy"^  are  to  be  introduced  as  positive 
values,  merely  with  the  restriction  that/^  be  greater  than  or  equal  to/,  in 
either  case. 


3.    RELATIONS    BETWEEN    THE   PRIMARY   AND   SECONDARY 

FOCAL   PLANES. 

Since  F^  and  F.^  have  been  shown  to  be  dependent  upon  /j,  /,,  and  /',  it 
is  evident  that,  for  fixed  values  of  f^  and  f.,^  the  resultant  foci  will  be  ren- 
dered dependent  entirely  upon  whatever  value  may  be  given  to  the  angle  y. 

It  is  further  obvious  that  the  refraction  of  one  cylinder  will  be  affected 


CONGENERIC   CYLINDERS.  63 

most  by  the  other  when  their  axes  coincide,  or  when  ;'  =  0°,  and  least 
when  their  axes  are  at  right  angles  to  each  other,  or  when  ;'  =  90°. 

We  shall,  consequently,  fix  upon  the  limits  of  Fi  and  F.^  for  these  ex- 
tremes of  y. 

Introducing  ^  =  0°,  and  consequently  cos  2^  =  +  1,  into  the  formulae 
(30)  and  (33;,  we  obtain,  for/,  >  /„ 

rr  f\fi  J\Ji 


V  4  [y;  (/,+/.)  +  (/■  +  /.)  a  -/■  -/J]     ^ 

For  F,  =  f{\^  we  shall  have  as  the  refraction 

-=:  =  —  +  -p?  consequently, 

4.  When  the  axes  of  the  congeneric  cylinders  coincide,  the 
primary  focal  plane  will  correspond  to  that  focal  plane  which 
is  defined  by  the  smn  of  the  refractions  of  the  cyhnders, 
whereas  the  secondary  focal  plane  will  be  at  infinity. 

This  is  shown  in  Plate  II,  Fig.  1 . 

Introducing  y  =  90°,  and  consequently  cos  2r  =  cos  180°  =  —  1, 
into  (30)  and  (33),  we  have,  for/  >/, 

C  J\Ji  JlJ'i    f 


p J  I  Ji fxJ'l   f 

_  .-.     F,:F,  =./,:/, (35) 

*  00  The  sign  for  infinity. 


64  DIOPTRIC   FORMULA. 

As/j  and/2  correBpond  to  the  positions  of  the  elementary  planes  E^  and 
E^^  it  follows  that 

5,  The  primary  and  secondary  focal  planes  coincide  with, 
their  correlative  elementary  focal  planes,  when  the  axes  of  the 
congeneric  cylinders  of  unequal  refraction  are  at  right  angles 
to  each  other. 

This  is  demonstrated  in  Plate  II,  Fig.  2. 

In  the  same  relation  (35),  if/  =  f.^^  then  F^  =  F^,  or 


6.  The  primary,  secondary,  and  elementary  focal  planes  all 
merge  into  one  plane,  when  the  axes  of  the  congeneric  cylin- 
ders of  equal  refraction  are  at  right  angles  to  each  other. 

As  in  this  case  we  have  but  one  focal  plane,  the  refraction  corresponds 
to  that  of  a  spherical  lens. 

F^  being  chosen  to  signify  the  primary  focal  distance,  it  will  have  to  be 
less  than  F.^,  yet  if/  >  /^,  we  should  find,  in  consequence  of  the  relation 
(35),  that  F^  >  /^,.  To  retain  the  significances  of  F^  and  F,,  it  will  there- 
fore be  necessary  to  substitute/  by  the  greater  given  value  of  cylindrical 
focus,  and/  by  the  lesser,  as  stated  under  the  formulae,  page  62. 

Owing  to  the  previous  considerations;  between  the  limits  of  0°  and  90° 

for  Y,  we  are  then  to  conclude  that  /\  will  vary  between    /■    "j  r  and  /, 

while  F^  varies  between  00  and  /,  as  the  nearest  and  most  remote  limits 
of  focal  distance  for  F^  and  F.^^  respectively. 

As  an  illustration,  let  Fig.  1,  Plate  II,  represent  two  combined  convex 
cyhnders  of  unequal  refraction,  with  their  axes  coincident,  and  so  united 
as  to  permit  of  the  rotation  of  one  of  the  cylinders  upon  the  true  planes  of 
their  faces,  about  the  optical  center  0. 


CONGENERIC   CYLINDERS.  66 

In  the  position  shown  (f  =  0°),  the  shortest  possible  focal  distance  /^j  of 
the  primary  focal  line  will  be   >  '  '•'   ,  which  corresponas  to  the  combined 

refraction,  ---\-   .  ,  of  the  cylinders  in  the  active  plane.      In  the  secondary 
y  1     y  2 

plane,  F.,=(x>;  consequently,  ~e^  =  —  =  0,  which  corresponds  to  the  re- 
fraction in  the  axial  or  passive  plane  of  the  cylinders. 

The  slightest  change  in  the  position  of  one  of  the  cylindrical  axes  will 
give  rise  to  a  definite  value  of  the  angle  y  in  the  Formula  III,  thereby 
bringing  F.^  within  the  limits  of  finite  distance,  while  decreasing  the  value 
of  F^  in  the  Formula  II. 

For  each  successive  increase  in  the  angle  r,  the  primary  focal  plane  cor- 
responding to  F^,  will  recede  farther  and  farther  from  the  combined  lenses 
towards  F^,  while  the  secondary  focal  plane,  corresponding  to  F.^,  ap- 
proaches nearer  and  nearer  from  oo  to  F.,,  until  ^  =  90°,  when  /^j  will  have 
reached  F^  on  the  moment  that  F,  merges  into  F.,,  as  shown  in  Plate  II, 
Fig.  2. 

Rotation  of  one  of  the  cylinders  is  thus  associated  with  corresponding 
changes  in  the  distances  F^  and  /^,,  while  the  movements  of  their  correla- 
tive focal  planes  will  be  in  opposite  directions  to  each  other ;  which  proves 
that : 


7.  The  primary  and  secondary  focal  planes  are  conjugate 
planes,  subject  to  variations  of  the  angle  between  the  axes  of 
the  congeneric  cylinders. 

In  order  to  comply  with  this  law,  in  constructing  the  Plate  I,  it  has  been 
necessary  to  select  elementary  foci  in  marked  disproportion  to  the  curva- 
tures of  the  cylinders  ;  otherwise  the  secondary  focus  F,  could  not  be  brought 
within  the  space  allotted  for  the  diagram. 


DIOPTRIC    FORMUL/E 

FOR  COMBINED 

CONTRA-GENERIC   CYLINDERS 


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II.  DIOPTRIC  FORMUL/E 

FOR  COMBINED 

CONTRA-GENERIC  CYLINDERS 


I.    RELATIVE    POSITIONS  OF   THE  PRINCIPAL    POSITIVE   AND 
NEGATIVE  PLANES  OF  REFRACTION. 

In  a  combination  of  convex  and  concave  cylinders,  we  can  no  longer  have 
the  primary  and  secondary  planes,  which  we  have  learned  to  consider  as 
planes  of  greatest  and  least  refraction,  but,  instead,  we  shall  have  a  plane 
of  greatest  positive  and  one  of  greatest  negative  refraction,  synonymouslj' 
with  the  generally-adopted  distinction  between  convex  and  concave  lenses, 
as  designated  by  the  signs  +  (plus)  and  —  (minus),  respectively.  As  the 
refractions  of  the  convex  and  concave  elements  in  the  combination  are  op- 
posing forces,  the  plane  of  greatest  positive  refraction  will  evidently  lie  be- 
tween the  active  plane  of  the  convex  and  the  axial  plane  of  the  concave 
cylinder,  whereas  the  plane  of  greatest  negative  refraction  will  be  between 
the  active  plane  of  the  concave  and  the  axial  plane  of  the  convex  cylinder. 

In  Plate  III,  therefore,  the  plane  DD^o^o  of  greatest  positive  refraction 
is  shown  between  c  and  A^  and  the  plane  dd^o^o  of  greatest  negative  refrac- 
tion between  C  and  a.  These  planes,  being  at  right  angles  to  each  other, 
divide  each  of  the  angles  A^o^c^  and  C^o^a^  into  «  and  {i. 

To  establish  the  formulae  for  combined  contra-generic  cylinders,  we  shall 

therefore  have  to  ascribe  another  significance  to  the  angles  a  and  /3. 

69 


70  DIOPTRIC   FORMULAE. 

The  deviation  of  the  axes  Aoa  is  equal  to  angle  A^o^a^  =  y,  and,  oince  c^o^ 
is  perpendicular  to  a^o^,  a  +  /?  +  ^  is  equal  to  90° ;  consequently, 

a  +  /?  =  90°  —  r (36) 

The  elementary  focal  planes  E^  and  ifj,  corresponding  to  the  focal  dis- 
tances f^  and/j,  respectively,  are  exhibited  on  opposite  sides  of  the  com- 
bined cylinders ;  since  E^,  for  the  concave  cylinder,  is  virtual  and  in  the 
negative  region  before  the  lens,  whereas  E^,  for  the  convex  cylinder,  is  in 
the  positive  region  behind  the  lens.  Consequently,  for  the  point  Z>,  the 
convex  cylinder  c  contributes  as  its  greatest  amplitude  of  deflection  D^Z^, 
perpendicular  to  a^o^  in  the  plane  Ey  The  greatest  amplitude  of  deflection 
for  the  concave  cylinder  C  is  D^  V^,  perpendicular  to  A^o^  in  the  virtual 
plane  E,^.  As  the  incident  ray  at  Z)  will  be  refracted  by  the  concave  cylin- 
der, as  if  emanating  from  a  correlative  point  F^  of  the  virtual  axial  line 
V^Oq,  it  is  evident  that  the  direction  of  the  ray  refracted  by  it  will  be  V^D  V^ 
The  proportionate  deflection  contributed  by  the  concave  cylinder,  measured 
in  the  plane  E^  will  consequently  be  D^  Vy 

Provided  the  point  D  is  properly  chosen,  it  will  be  a  point  of  the  plane 
of  greatest  positive  refraction,  that  is  to  say,  when  the  resultant  deflec- 
tion D^M^^  accruing  from  the  associated  deflections  D^  V^  and  D^Z^  in  the 
parallelogram  of  forces  D^  V^M^Z^,  is  directed  towards  the  optical  axis. 

To  insure  D^M^  being  so  directed,  it  is  obvious  that  the  associated  de- 
flections, Z>jZj  and  D^  V^  must  also  be  measured  in  the  plane  E^,  in  the 
positive  region  behind  the  lens. 

Similar  reasoning  will  apply  to  the  point  d  as  being  in  the  plane  dd^o^o 
of  greatest  negative  refraction.  In  this  instance  d^m^  being  a  force  di- 
rected from  the  optical  axis,  in  the  plane  E^^  is  to  be  taken  negative,  synon- 
ymously with  the  plane  of  greatest  negative  refraction. 

The  relations  between  a  and  /?  are  to  be  determined  by  an  analogous 
method  to  the  one  given  for  congeneric  cylinders,  whereby  we  obtain 

sin  2a  =  sin  2^3^.^ (37) 

as  defining  the  positions  of  the  planes  of  greatest  positive  and  negative  re- 
fraction, which  are  again  at  right  angles  to  each  other. 


CONTRA-GENERIC    CYLINDERS.  71 

We  here  also  find  the  sines  of  the  angles,  2a  and  2/?,  to  differ  by  the  co- 

f  90° V 

efficient  —-.     Hence,  wheny^=y],  we  shall  have  «  =  /J^: — -^  or, 

/o  ^ 

8.  For  combined  contra-generic  cylinders  of  equal  refraction, 
the  plane  of  greatest  positive  refraction  equally  divides  the  an- 
gle between  the  active  plane  of  the  convex  and  the  axial  plane 
of  the  concave  cylinder ;  and  the  plane  of  greatest  negative  re- 
fraction similarly  divides  the  angle  between  the  active  plane  of 
the  concave  and  the  axial  plane  of  the  convex  cy Under. 

In  case/o  >/„  then  ;5  >  « ;  or, 

9.  When  the  convex  cylinder  is  stronger  than  the  concave 
cyUnder,  the  plane  of  greatest  positive  refraction  will  be  nearer 
to  the  active  plane  of  the  convex,  while  the  plane  of  greatest 
negative  refraction  will  be  proportionately  farther  from  the  ac- 
tive plane  of  the  concave  cyhnder. 

In  case/,  >  /,  then  «  >  /3 ;  or, 

10.  When  the  concave  cylinder  is  stronger  than  the  convex 
cyhnder,  the  plane  of  greatest  negative  refraction  will  be 
nearer  to  the  active  plane  of  the  concave,  while  the  plane  of 
greatest  positive  refraction  will  be  proportionately  farther  from 
the  active  plane  of  the  convex  cylinder. 

This  is  manifest  in  the  diagram. 

The  values  of  a  and  /?  may  be  expressed  in  terms  of/,,  /„  and  ?'  in  a  sim- 
ilar manner  to  that  shown  in  the  previous  theorem,  when  it  can  be  shown 
that. 


cos 


V2^2 


/o  — /,  cos  lr 


2  Vf^-yj,  cos  2r  +// 


(VI) 


This  and  the  transposed  equation  (36),  /?  =  90°  —  (^y  -\-  a)^  suffice 
to  locate  the  positions  of  the  principal  planes  of  refraction  ;  the  angle  a  being 
counted  from  the  axis  of  the  convex  cylinder. 


72  DIOPTRIC   FORMULAE. 

2.     POSITIONS    OF    THE    POSITIVE    AND    NEGATIVE     FOCAL 

PLANES. 

The  positions  of  the  positive  and  negative  focal  planes  will  evidently  here 
also  be  determined  by  the  resultant  rays,  DM^  and  dm^,  and  their  correla- 
tive intersections  with  the  optical  axis  at  O^  and  0„. 

O,  Wj  will  therefore  represent  one-half  the  focal  line  in  the  positive  region 
behind  the  lenses,  and  O^M^  one-half  the  virtual  focal  line  in  the  negative 
region  before  the  same. 

The  elHpses  shown  in  the  planes  E^  and  E^^  are  of  the  same  significance 
in  this  as  in  the  preceding  combination. 

In  the  plane  of  greatest  positive  refraction,  Z>Z>,  YO^,  we  have 
DY.DD,^  YO^  -.D.M,. 

Substituting,  DY  =  O^o  =  F,     as  the  positive  focal  distance ; 

YO^  =  Do  =-  radius  =  1. 

••■^.=    7^ <''^' 

In  the  parallelogram  D^  V^M^Z^,  the  angle  between  the  forces,  D^  F,  and 
Z>jZj,  is  equal  to  180°  —  y,  since  D^Zy  :  Z^o^,  and  D^  V^      A^Oy 

•••  A^i  =V  (,D,Z,f+{D,V,y+2  iD.ZJ  (D,V,) cos (180''— r). 
In  the  oblique  plane  D^Vf^DV^Dj,  we  find, 

Z>o  V^  =  sin  <^  D^o^Aq  =  sin  <^  Dfi^A^  =  sin  /S. 


CONTRA-GENBRIC     CYLINDERS.  73 

.-.    D,l\  =-^J.  siny5. 

/^,Z,  =  sin  (<J^  ^'xOxC^  —  <^  A^i^i)  ~  '^"^  (^^°  —  "■)  —  cos  «. 

Substituting  these  values  in  tlie  equation  for  Z>,  yT/,,  equation  (47)  be- 
comes, 


F,  = 


I  r -i 7 ' 

A  /  cos^ ""'"(/)  ^^^^  '^ — ^/  ^^^  ^  c®^ "  c®^  y 


and  which,  through  equation  (36),  may  be  given  the  form  : 

F  =^ 

I'^i L/i  (/.  -/o  cos  2r)  +  (/o  -/)C/o  +  T//o"^-2/o/cos2r+/,0]' 

(48) 

Substituting,  cos  2^  ^=  1  —  2  sin^ ;', 


-^.p_  ^  -^'-^° .    .  .   (VII) 

^iJ±-Ql  +/o/8inV+  (/o-/x)  ^^Az:^^  +/o/.sinV. 

This  formula,  when  reduced  for  cylinders  of  equal  positive  and  negative 
refraction, /o  being  equal  to/,  =/,  assumes  the  simple  form 

F,  =  -^ (IX) 

^       sin  /-  ^      ^ 

In  the  plane  of  greatest  negative  refraction,  d^m^dO^X,  we  obtain, 

dX  :  dd^  =  XO^  :  d^m^. 

Substituting,         dX  =  0,p  =  —  /%,   as  the  negative  focal  distance; 

XO^^  =  do  =  radius       1. 

.-.     _/;  =  —  /-; (49) 

since  d^m^  is  to  be  taken  negative. 


74  DIOPTRIC   FORMULiE. 

In  the  parallelogram  d{L\m^z^^  the  angle  between  the  forces,  d^v^  and  fl?,^,, 
is  again  180°  —  y\  hence, 

d^^^  =  V  {d^2j  +  {d,vy  +  2  {d^z;)  {d,V^)  COS  (180°  —r)- 

In  the  oblique  plane  d^v^dv^d^,  we  find, 

d^v^  :  dd^  -—  d^^v^  :  dd^. 
d,v^  =  sin  «  Z>„<7o4  —  <  AMo)  =  sin  (90°  —  <  D^o^A,) 

=  sin  (90°  —  ;5)  =  cos  fi. 
dd^  =/u. 

.'.     flfjZ'j  =  "—  cos  /?. 

d^z^  =  sin  <^  ^i<?i2'i  =  sin  «. 
Substituting  these  values  in  the  equations  for  d^}n^  and  (49),  we  have, 

_F - /■ 

■'  0  ■ —  , J- ^2 7 

-»  /  sin^  «  +    (  ^  )  COS"  /?  —  2  •— }-  sin  «  cos  ^5  cos  ^ 

and  which,  by  the  aid  of  equation  (36)  maj^  be  written,  —  Z^,  = 

fj. 

^l[/;(7i-/ocos  2r)  +  (/o-/J  U-  l//„=^-2/o/,  COS  2rT7?)] 

(50) 

/x/o 

(VIII) 

which  differs  from  the  formula  given  for  F^  merely  by  a  transposition  of 
the  elements  in  the  factor  before  the  second  radical,  and,  consequently, 
when  reduced  to  cylinders  of  equal  refraction,  also  becomes 


CONTRA-GENERIC     CYLINDERS.  75 

-/^,  =  -   A (X) 

sin  /' 

The  formulae  (IX)  and  (X)  correspond  to  those  which  were  appUed  to 
the  Stokes  Lens. 

In  reducing  the  preceding  formula)  for  given  values  of  cylindrical  foci, 
f^  is  to  be  substituted  by  the  focus  of  the  concave,  and  f^  by  the  focus  of 
the  convex  cylinder,  both  being  introduced  as  positive  values. 


3.    RELATIONS    BETWEEN    THE    POSITIVE    AND    NEGATIVE 

FOCAL  PLANES. 

As  in  this  combination  the  cylinders  likewise  affect  each  other  most 
when  their  axes  coincide,  and  least  when  their  axes  are  diametrically 
opposed,  we  may  here  also  fix  upon  the  limits  of  F^  and  —  F^  for  >'  =  0^ 
and  Y  =  90°,  as  in  the  previous  theorem. 

When  Y  =  0°,  or  cos  2^'  =  4-  1,  from  the  equations  (48)  and  (50)  we 
find,  for/„  >  /;, 

■p /  1  /  0 ' JiJo 

'  ~  VK-A  i/o-A)  +  (/o-/i)  (/o  +/o-/x)]  ~/o  -/ 

~  Vii^A  (Ao-A)  +  (fo-A)  (/o-/o+y;)]^     o^=~  ^• 

.-.  F,:-F,  =  /^:-  ^. (51) 

Jo  J  \ 

For  F^  =  >        x»  we  have  as  the  refraction ->,  =  y y-;  consequently, 

Jo        /l  ^ \  J  \  Jo 

11.  When  the  convex  cylinder  is  of  greater  refraction  than 
the  concave,  and  their  axes  are  coincident,  the  positive  focal 
plane  will  coincide  with  that  focal  plane  which  is  defined  by 
the  difference  of  the  refractions  of  the  cylinders,*  whereas  the 
negative  focal  plane  will  be  at  infinity. 

*Or  the  sum  of  their  refractions  when  taken  as  positive  and  negative  elements. 


—  F. 


76  DIOPTRIC  formul.t:. 

Placing  Y  =  0°,  or  cos  2^  =  +  1,  in  the  equations  (48)  and  (50),  we 
have,  for/,  >/„, 

-/i/o fi/o  „ 


F,= 


-Fo  = 


V\  [/  (/i  -/o)  -  (  /,  -  /o)  (/o  +  /,  -/o)]  0 

/i/o yi/o 


!/*[/:  (/.  -/o)  -  (/  -/o)  (/o  -/  +  /o)3  /.  -  /« 

/i        /o 

For  —   /^)  =   —  //'    ^'®    have    as   the  refraction 
A      /o 


^  =  —  \^ -2- J    consequently. 


12.  Wtien  the  concave  cylinder  is  of  greater  refraction  than 
the  convex,  and  their  axes  are  coincident,  the  negative  focal 
plane  will  coincide  with  that  focal  plane  which  is  defined  by 
the  difference  of  the  refractions  of  the  cylinders, ^'^  whereas  the 
positive  focal  plane  will  be  at  infinity. 

This  is  shown  in  Plate  IV,  Fig.  1. 

Introducing  y  =  90°,  or  cos  ly  =  cos  180°  =  —  1  in  the  equations  (48) 
and  (50),  we  have,  for/  ^/j, 

"^"^  ~  l/K/  (/  +/o)  +  (/o  -/)  (f.-fo  -/,)]^~    /x'  "^  "  ^" 

.-.  F,  ■.-]■,=./,: -I, (53) 

•  Or  the  Bum  of  their  refractions  when  taken  as  positive  and  negative  elements. 


CONTRA-GENERIC     CYLINDERS. 

From  which  we  deduce  : 


13.  The  positive  and  negative  focal  planes  coincide  with  their 
correlative  elementary  focal  planes,  when  the  axes  of  the  contra- 
generic  cyhnders  are  at  right  angles  to  each  other. 

This  is  demonstrated  in  Plate  IV,  Fig.  2. 

Between  the  hmits  of  0°  and  90°,  fory„  >/,,  we  have  consequently 
found  F^  to  vary  between  the  limits  of  r        r  ^.nd  /j  behind  the  combined 

Jo        J I 

lenses,  while  F^  varies  between  the  limits  of  ooand/^  on  the  incident  side 
of  the  same. 

The  convex  being  stronger  than  the  concave  cylinder,  it  is  evident  when 
their  axes  coincide  that  their  combined  refraction  will  be  equal  to  that  of  a 

periscopic  convex  cylinder,  since  ~  =z  ~ T  ^"^  ^^^  active  plane ;  and 

^\  J\  Jo 

-^  =      -  =  0  in  the  passive  plane. 

/;      oo 

Between  the  same  limits,  when/j  >y'o,  F^  will  vary  between   /    'S  and 

J I        Jo 

/„  on  the  incident  side  of  the  combined  cylinders,  while  F^  varies  between 
00  and/i  behind  the  same.     (See  Plate  IV.) 

In  this  case,  when  the  axes  coincide,  it  is  evident  that  the  resulant 
refraction  will  be  equal  to  that  of  a  periscopic  concave  cylinder,  since 

^  =  —  (^ J- )  in  the  active  plane ;   and  -j^  =  — ^  =  0  in  the 

Fo  V/o         /i  -^  F^       CO 

axial  plane. 

Therefore,  with  an  inequality  in  the  refractive  powers  of  the  cylinders, 
rotation  of  one  of  them,  from  0°  to  90°,  will  be  associated  with  correspond- 
ing changes  in  the  position  of  the  resultant  focal  planes,  between  the 
limits  of  infinity  and  the  focus  of  the  weaker  cylinder  on  the  one  side, 
and  between  that  focal  plane  which  corresponds  to  the  difference  of  their 
refractions  and  the  focus  of  the  stronger  cylinder  on  the  other.  Since  in 
this  case  the  approach  of  one  focal  plane  is  accompanied  by  a  corresponding 


78  DIOPTRIC  FORMULiE. 

recession  of  the  other  on  the  opposite  side  of  the  lenses,  their  movements 
are,  as  in  the  previous  theorem,  in  opposite  directions. 

When  the  cylinders  are  of  equal  refractive  j^ower,  f^  being  equal  to  /„,  it 
follows,  from  the  relation  (53),  that  F^  =  F^,  so  that  between  the  hmits  of 
0"  and  90°,  F^  will  vary  between  infinity  and/^  on  the  positive  side,  while 
Fq  varies  between  infinity  and  /„  on  the  negative  or  incident  side  of  the 
combined  cylinders. 

Consequently,  when  the  axes  coincide,  +  Fi=  -r  oo  and  —  -^o  =  — 
00  .  This  is  evident,  since  the  refractions  of  equal  convex  and  concave 
cyhnders,  under  such  circumstances,  neutralize  each  other  throughout. 

By  the  previous  considerations  we  therefore  here  also  find  : 


14.  Tlie  positive  and  negative  focal  planes  are  conjugate 
planes,  subject  to  variations  of  the  angle  between  the  axes  of 
the  contra-generic  cylinders. 

The  diagram,  Plate  III,  has  been  constructed  in  accordance  with  the 
foregoing  provisions. 

For  practical  purposes,  it  will  be  found  more  convenient  to  use  the 
formula  in  the  next  chapter. 


DIOPTRAL    FORMUL/E 

FOR  COMBINED 

CYLINDRICAL  LENSES 


PLATE  ir 


III.     DIOPTRAL*    FORMUL/E 

FOR    COMBINED 

CYLINDRICAL  LENSES. 


I.    RELATION  BETWEEN  THE  PRINCIPAL  PLANES  OF  REFRAC- 
TION AND  THE  REFRACTIVE   POWERS 
OF  THE   CYLINDERS. 

As  the  task  of  reducing  diophies  to  their  focal  distances  would  render 
calculation  by  the  preceding  formulae  somewhat  arduous,  we  may  here 
give  the  formulae,  expressed  in  refraction,  which  will  be  found  especially 
convenient  when  applied  to  combinations  of  the  metric  system. 

Since  original  publication,  these  formulae  have  been  given  their  simplest 
possible  fomi.  The  new  formula?,  IIZ?,  IIIZ?,  VIIZ?  and  VIIIZ?  are  now- 
introduced  as  sequences  to  the  original  formulae,  which  are  also  given,  and 
whose  transformations  have  been  accomplished  through  convenient  substi- 
tutions from  the  equations  54a,  54  and  55. 

For  the  focal  distance  F.  we  have  as  the  refraction    ^  =  /?„  and  for/, 

and  /,  similarly,  -^   =  r   and  -j  =  n,  which  designate   the   dioptral 
powers  of  the  cylinders. 

By  these,  and  similar  substitutions  for  other  foci,  we  may  give  the  pre- 
ceding formulae  (I)  to  (^X),  the  following  form  : 


*The  choice  of  this  adjective  would  seem  justifiable,  since  the  unit  "dloptry  "  has  been  chosen  in 
distinction  to  "dioptric,"  which,  though  rclnti-d,  hiis  auother  significance. 

«1 


82  DIOPTRAL    FORMUL.«. 

THE    DIOPTRAL   FORMULAE   FOR   COMBINED 
CONGENERIC    CYLINDERS. 


cos 


a  =       A  -f  i  r,  +  r^  cos  2r 

y^       2  i/rj2_|-2r,r,  cos2r+  r/ 


=  i  (''i  +  ^2  +  I'V,  +  ''z)'  — 4r,r,  sin 2  J-). (IID) 

^2  =  V"!  (r,  +  r,)'  —  r,r,  sin  V  —  {r,  -\-  r,)  V^i^r,  +  r,f  —  r,r,  sin'T- 

=  i  C'',  +  ''.  —  l^Vi  +  ^-2)'  —  4^1  n  sirT^. (IIIZ?) 

To  retain  the  significances  of  R^  and  J^.,,  in  calculating,  rj  should  repre- 
sent the  greater  cylindrical  refraction. 

When  the  cylinders  are  of  equal  power,  then  r,  =  r.^  =  r,  so  that 

^^  =  r  (1  +  cos  r) (IVZ^) 

/?,  =  r  (1  —  cos  r) cvz;) 


THE    DIOPTRAL    FORMULA    FOR    COMBINED    CONTRA- 
GENERIC    CYLINDERS. 


_       /^  _L  ^  r^  _  y-p  COS  2r 

y^r^  COS  2r  -f  ^' 


-^«=  V2+2  7r^=^ir=^^=^'-  •  •  •  ^'^'''^ 


^1  =  V\irx—'^J'  +  ^%sinV  +  {r^  —  r„)  l/i(^i  — O'  +  ^^o  sin^*  r- 


iC'-i  — ^»+  l-^C'-i  — ^oy  +  4r,r„sinv) (VIIi9 


—  ^0  =  —  1/*  (r J  —  r„)'  +  rj  r^  sinV  +  (^o  —  ^,)  1 ' i  (^,  —  ^o)'  +  ''i ^0  sinV- 


=  i(^i-^o--l/(^,— '-o)'  +  4r,r„sinV) (VIIIZ;) 


CYLINDRICAL    LENSES.  83 

when  tlio  cylinders  are  of  equal  power,  then  r^  z=  r^^.  r^  henco 

R^  =  rQmr (IXZ?) 

—  R^  =  —  r  sin  y (XD) 

If,  in  (IID)  and  (IIIZ>),  the  convex  element  r,  be  replaced  by  the 
concave  element  —  r^,  we  obtain  (VIID)  and  (VIIIZ>). 

By  the  aid  of  the  preceding  formulse  we  may  also  arrive  at  the  following 
signiticant  facts. 

The  formula)  dlD)  and  (lllV)  may  be  written  : 

R;'  =  ^(r,  -i   r,y^r,r,  sin'^  r  +  (r,  +  r,)  l/j  (r^  +  rj' —  r,r,  sin' r), 


/?/  =  ^  (r,  +  rj  —  r^r.^  sin'  ^  —  (r^  +  r,)  Vi(r^  -f-  r  J'  —  r,;%  sin*  r, 

which,  by  addition,  result  in  the  equation, 

/?,  -  +  R,  -  ^  (r J  +  rj'  —  2^1?-,  sin  V- 

.  •.     (R,  +  ^.)-^  -  2^,  /?,  =  (r ,  +  r,)^  _  2r^r,  sin  -'  y 
. • .     (^j  +  R,r  =  (r^  +  r,y  —  2r^r ,  sin* r  +  2/?^/?,. 

Multiplying  (IIZ>)  by  (III/?),  we  find, 

2^?,/?.,  =  2rjr,  sin-'r (54a) 

.-.      R^-^  R,^^r,-{-r., (54; 

From  which  we  conclude  : 


15.  The  Slim  of  the  primary  and  secondary  refractions  is  a 
constant,  being  equal  to  the  sum  of  the  elementary  refractions 
for  any  combination,  and  all  deviations  of  the  axes  of  two 
combined  congeneric  cylinders. 

In  the  same  manner,  we  obtain  from  the  formula)  (VIIZ>)  and  (VIIIZ?), 
R^  —  R^^r,  —  r„ (65) 


84  DIOPTRAL   FORMULAE. 

and  therefore  here  also  find, 

16.  The  sum  of  the  principal  positive  and  negative  refrac- 
tions is  a  constant,  being  equal  to  the  sum  of  the  positive 
and  negative  elementary  refractions  for  any  combination, 
and  all  deviations  of  the  axes  of  two  combined  contra-generic 
cylinders. 

Ab  the  total  inherent  refraction  always  remains  the  same  for  any  combi- 
nation, the  angle  y  merely  performs  the  function  of  allotting  the  proportions 
of  refraction  R^  and  R^^  or  R^  and  R^^  in  the  resultant  principal  planes. 

By  the  equations  (54)  and  (55),  calculation  may  be  greatly  simplified. 
R^  being  determined  for  a  specific  value  of  r,  we  may  readily  determine  R^ 
and  R^^  by  transforming  these  equations,  as  follows  : 

R.  =  r,^r,-R, 

—  R^=r,  —  r^  —  Ry 

This  is  demonstrated  in  the  appended  tables,  although  it  has  not  been 
utilized  in  calculating ;  on  the  contrary,  a  study  of  these  led  to  the 
above  deductions. 


SPHERO-CYLINDRICAL  EQUIVALENCE 


PLATE  IV 


THE  REFRACTION   BY  COMBINED 

CONTRA-GENERIC  CYLINDRICAL  LENSES 


Fig.  1 


Fig.  2 


^-^F.-^h 


:' :  p  =  J  "J  'i 
'  J  .'1      Jo 


CHAS.   F.   PRENTICE 

COPYRIGHT,    1988 


IV.    SPHERO-CYLINDRICAL  EQUIVALENCE. 


Since,  for  any  combination  of  cylinders,  the  principal  planes  of  refrac- 
tion are  at  right  angles  to  each  other  for  all  values  of  y^  there  can  be  no 
reasonable  doubt,  owing  to  the  provisions  made  at  the  opening  of  this 
demonstration,  as  to  the  equivalence  of  a  sphero-cylindrical  lens  to  one 
composed  of  combined  cylinders.  However,  as  the  use  of  such  lenses  is 
at  present  confined  to  the  correction  of  errors  of  refraction  in  the  human 
eye,  it  is  evident,  from  the  movements  of  the  eye  behind  the  fixed  lens, 
that  the  visual  axis  cannot  at  all  times  coincide  with  the  optical  axis  of 
the  lens  chosen  ;  therefore,  in  those  instances  where  substitution  of  one 
form  of  lens  for  the  other  proves  to  be  unsatisfactory,  the  cause  might 
seemingly  be  explained  by  a  possible  difference  becoming  manifest  for  the 
more  peripheral  incident  rays,  though  these  be  equally  distant  from  the 
optical  center  of  each  lens.  In  other  words,  the  available  field  in  the  one 
may  be  greater  or  less  than  in  the  other  ;  yet  even  this  would  probably  only 
be  appreciable  in  lenses  of  extreme  curvature,  and  possibly  in  combinations 
where  the  cyhnders  differ  widely  in  power.  Ht)wever,  this  would  remain 
to  be  shown. 

To  substitute  a  sphero-cylindrical  lens  for  combined  cylinders  is  a 
proposition  which  merely  demands  that  the  focal  interval  should  be  the 
same,  at  the  same  distance  from  the  principal  plane,  at  the  optical  center, 
for  each  of  the  compound  lenses.  The  distances  F  ^  and  F.^  being  deter- 
mined for  any  angular  deviation  r  of  the  axes,  in  a  combination  of 
congeneric   cylinders,  for  instance,  the  substitution  is  accomplished  by 

making  a  sphero-cylindrical  lens  in  which   the  focus  of  the  spherical 

F  F 
element  is  equal  to  /%  ,  and  of  the  cylindrical  element  is  equal  to       '    '      or, 

i^i — i"\ 

if  expressed  by  refraction,  = -=r  sph.  =  -=:  cyl.  =  — -. 

87 


88  SPHERO-CYLINDRICAL     EQUIVALENCE. 

Should  it  be  desired  to  place  the  primary  and  secondary  planes  of  the 
sphero-cylindrical  lens  so  as  to  coincide  with  those  resulting  from  a  combi- 
nation of  two  definitely  placed  congeneric  cylinders,  it  will  be  necessary  to 
refer  to  the  formula  (I)  and  to  the  laws  2  and  3. 

Comparing  the  sphero-cylindrical  equivalent  with  its  corresponding 
rotating  cylinders,  reference  being  had  to  Plate  II,  Figs.  2  and  3,  we  find 
a  decrease  in  the  angle  y  from  90°  to  zero  to  effect  a  corresponding  de- 
crease in  the  spherical  element  F.^,  from  the  focus/,  to  oo;  this  being  asso- 
ciated  with   a   cylindrical   element  of    the  focus   Fc,    which  constantly 

increases  from  the  focus  /^  \  to  ^^1  .  In  other  words,  a  gradually 
decreasing  potency  of  the  spherical  refraction  —=:  from  — .  to  —  =0,  gives 
way  to  a  proportionately  increasing  cylindrical  refraction  — ,  from  -z. 

re  J I 

—  to  --^  -f-  — ^  .     As  an  instance,    if/.  =  /  =  /  ^r  will  increase  from 
A        A        Jl  '  ^<^ 

112 

— -r-  =  0  to    7-  ,  or  twice  the  refraction  of  either  cylinder.     In  this 

j\        Jl.  J 

case,  all  successive  values  of  cylindrical  refraction  will  therefore  be  inhe- 

2 
rent  between  0  and  -7-  . 

Should  a  means  be  devised  to  suppress  the  spherical  element  for  each 
successive  value  of  ^,  the  remaining  varying  cylindrical  element  being 
thus  rendered  available  for  measuring  corresponding  degrees  of  astigma- 
tism in  the  eye,  the  formulae  here  advanced  would  prove  of  service  in  ob- 
taining the  graduations  upon  the  rotating  scale  of  such  an  instrument. 

While  there  are  few  cases  of  astigmation  which  demand  correction  by 
combined  cylinders,  we  may  nevertheless  be  permitted  to  passingly  allude 
to  certain  methods  of  procedure  in  such  instances.  We  shall  confine  the 
subject  to  congeneric  cylinders.  In  a  case  of  astigmatism  which  has  been 
found  to  be  corrected  by  two  cylinders  combined  under  the  angle  /',  the 
lenses  should  be  withdrawn  from  the  trial  frame,  when  they  are  to  be 
superposed  with  their  plane  surfaces  in  contact;  and  in  such  manner  as  to 
facilitate  their  being  rigidly  held  in  the  required  position  for  y. 

The  positions  of  the  principal  planes  of  refraction  may  then  be  estimated 
for  this  fixed  combination,  the  same  as  if  it  were  a  single  lens,  though 
without  regard  to  the  exact  nature  of  the  elements  constituting  it.     The 


SPHERO-CYLINDRICAL     EQUIVALENCE.  89 

powers  of  the  principal  planes  of  refraction  will  be  revealed  by  neutralizing 
with  the  lenses  from  the  trial  set.  The  spherical  and  cylindrical  elements 
thus  determined  are  then  to  be  substituted  in  the  trial  frame,  when  rota- 
tion of  the  cylinder  will  lead  to  that  position  of  it  which  produces  the  best 
acuteness  of  vision.  The  spherical  and  cylindrical  elements  will  probably 
then  also  bear  of  further  modification,  in  case  any  error  may  have  been 
made  at  the  outset.  In  lieu  of  this  practical  method,  recourse  must  be 
had  to  the  formula;. 

It  having  been  shown  that  successive  changes  in  the  angle  y  are  asso- 
ciated with  corresponding  changes  in  Fy  and  /v^,  the  above  substitution 
would  indeed  seem  advisable,  since  the  present  appliances  for  grinding  bi- 
cylindrical  lenses  are  not  constructed  with  sufficient  precision  to  enable 
opticians  to  fix  the  relative  positions  of  the  cylinders  beyond  mere  approx- 
imation. 

As  an  illustration,  let  us  select  two  congeneric  cyUnders  of  equal  foci, 
say  20  inches,  combined  under  the  angle  y  =  60°.  Introducing  these 
values  in  the  formulae  (IV)  and  (V),  we  find, 

F  =  2Q  =        20        _ 

1  +  cos  60°        1  -f  0.5  ~  ' 

F  = ?-^_  =  -  J-^^^  =  40 

1  —  cos  60°         1  —  0.5 

We  then  obtain  the  cylindrical  refraction  ~=r  ,  for  the  desired  sphero- 
cylindrical equivalent,  from  the  equation, 


1.  _  Jl  =    1 

X         F~F~ 


(56) 

Substituting  herein  the  calculated  values  for  /-',  and  F.,  gives. 


_1 J_  _  i  _  _J_  _  i    .nearly) 

13.33       ^~~  Fc~  19.99  ~  20  '^'^^^^^y^- 

-=^=27^   being  the  spherical  element,  we  therefore  have  the  sphere- 


F,  -40 

cylindrical  equivalent, 


4^  'p^-  -  ^  "y^- 


90  SPHERO-CYLINDRICAL     EQUIVALENCE. 

as  an  available  substitute  for  the  cylindro-cylindrical  lens, 

^r^  cyl.  axis  0°  Q  ^^  cyl.  axis  60° 
20    -^  20    *^ 

without  regard  to  a  definite  position  of  these  lenses  before  the  eye. 

By  way  of  comparison,  allowing  the  optician  to  make  an  error  of  appa- 
rently 60  small  an  amount  as  2°,  in  producing  the  same  cylindro-cylindrical 
lens,  we  obtain,  by  introducing  ^  =  62°  in  the  same  formulse, 

^  20 20         _    20 

^  ""  1  +  cos  62°  ~  1  +  0.469  ~  1.47  ~"  ' 

P  _  20 20_  _  _^  _  37  70 

^^  ~  1  —  cos  62°  ~  1  —  0.47  "~  0.53  ~ 

Substituting  these  values  in  the  equation  (56),  we  have, 

_1 1 ^1 1_ 

13.61       37.73  ~  /^c~"  21.29  ' 

from  which  we  obtain  the  sphero-cylindrical  lens. 


37.73  ^      ^  21.29 

Had  the  optician  been  required  to  make  a  sphero-cylindrical  lens 
27r  sph.  Q  ^x  cyl.,  his  execution  of  it  presenting  such  discrepancies  as 

„        sph.  Q  cyl.,  would  certainly  be  rejected  as  being  unsatisfac- 

tory,  on  account  of  the  notable   difference  of   2.27  inches  in  the  focal 
distance  of  the  spherical  element. 

On  the  other  hand,  instances  are  likely  to  occur  in  which  it  will  be  im- 
possible, by  the  advanced  method  of  neutralization  to  accurately  arrive  at 
the  sphero-cylindrical  equivalent. 

Since  ^q  cyl.  axis  0°  C  20  ^^^^  ^^^^  ^^°  =  3773  ^P^^-  ^  21^  ^^^' 
we  should  evidently  be  unable  to  accurately  neutralize  such  spherical  and 
cylindrical  elements  by  any  of  the  lenses  from  the  trial  set. 

In  those  instances,  therefore,  where  satisfactory  neutralization  of  the 
principal  planes  of  refraction  in  a  pair  of  combined  cylinders  cannot  be 


SPHERO-CYLINDRICAL    EQUIVALENCE.  91 

attained,  the  cylindro-cylindrical  lens  will  have  to  be  chosen,  again  under 
the  proviso,  however,  of  a  faultless  mechanical  execution.  However,  as 
in  most  instances  a  sphero-cylindrical  equivalent  will  be  available, 
we  are  to  suspect  error  in  our  estimate  of  the  refraction  of  an 
eye  which  seems  to  demand  cyUnders  combined  under  acute  or 
obtuse  angles. 

The  following  is  a  case  in  point  : 

A  cylindro-cylindrical  lens  —  ^Ti  cyl.  axis  0°  C  —  j?)  cyl.  axis  70° 
had  been  prescribed  for  Mr.  G.  B.  0.,  of  New  York,  by  his  oculist  in 
Philadelphia,  in  1880-1.     With  this  lens  the  vision  =  -,   for  the  left  eye. 

In  this  instance  the  sphero-cylindrical  equivalent  was  obtained  as 
follows  : 

The  lenses  being  congeneric  concave  cylinders  of  equal  refraction,  by  the 
formulaj  (IV)  and  (V),  for/=  40  and  y  =  70°,  we  have, 

it  being  admissible  to  neglect  the  fractions  for  such  focal  distances. 

By  law  2,  we  find  the  position  of  the  cylindrical  axis  eq\ial  -^  =  35*, 
and  consequently  the  sphero-cylindrical  equivalent, 

—  gQ  sph.    O—  ^^  cyl.  axis  35°. 

This  lens  was  substituted  with  the  knowledge  and  to  the  entire  satisfacr 
tion  of  the  patient. 

It  is  therefore  obvious  that  the  meridian  (125°)  of  greatest  refraction  in 
the  eye  had  not  been  disclosed  by  the  oculist's  diagnosis. 

The  M'^eak  spherical  element  in  the  substituted  lens,  while  being  an 
appreciable  factor  to  the  patient,  might  easily  have  been  overlooked  by  the 
practitioner. 

In  similar  cases,  the  advanced  formulae  must  prove  of  value  in  fixing 
upon  the  true  state  of  the  refraction. 


VERIFICATION 

OF  THE 

DIOPTRIC  AND  DIOPTRAL  FORMULA 


V.  VERIFICATION  OF  THE  FORMULAE. 


In  the  following  tables,  the  Dioptric  and  Dioptral  FommlgB  have  been 
separately  applied  to  combinations  of  cylinders  of  the  inch  and  metric  sys- 
tems, respectively.  It  would  be  inadmissible  to  substitute  the  generally 
adopted  inch-system  equivalents  for  dioptrics,  in  calculating,  on  account  of 
frequent  repetitions  of  the  former  as  factors  in  the  dioptral  formula?,  which 
would  naturally  increase  the  neglected  differences  to  an  unwarrantable  de- 
gree. For  the  purpose  of  obtaining  reliable  results,  the  calculations  have 
been  carried  to  the  fifth  decimal  point  under  the  radicals.  The  angles  30°, 
45°,  and  60°  have  been  chosen  so  as  to  exhibit  appreciable  differences  in 
the  corresponding  resultant  refractions,  which  are  thereby  also  brought 
within  the  lens-series  of  the  inch  and  metric  systems.  The  elementary 
foci  and  refractions  have,  in  a  measure,  been  arbitrarily  selected,  it  being 
noticeable  that  the  secondary  refraction  will  generally  be  beyond  the  limits 
of  neutralization  for  combinations  of  weaker  cylinders  whose  axes  deviate 
less  than  30°. 

The  Approximates  given  for  refraction,  in  Table  1,  will  at  times  appear 
to  conflict  with  the  laws  15  and  16;  this,  however,  is  to  be  attributed  to 
changes  of  proportion  occasioned  by  the  adopted  substitutions. 

To  substantiate  the  resultant  refractions  given  in  the  tables,  through  the 
practical  test  of  neutralization,  the  cylindrical  axes  should  first  be  accu- 
rately determined,  and  their  deviation  effectively  maintained  while  the 
plane  surfaces  of  the  cylinders  are  kept  in  absolute  contact. 

In  holding  these  lenses  while  neutralizing,  great  care  should  be  exercised 

to  prevent  slipping,  as  the  slightest  variation  in  the  position  of  the  axes  will 

prove  misleading.     In  this  practical  experiment,  the  observer's  eye  will 

generally    fail    to    appreciate    the    neglect    of    fractions    made    necessary 

through  using  lenses  from  the  trial  case. 

95 


96  VEEIFICATION    OF    THE    FOBMTJL-SJ. 

In  erplanation  of  the  tables  (1)  on  the  following  page,  under  the  caption 
"Elementary  Foci,"  are  given,  for  instance,  the  foci  (/j  =  16  inches,  and 
/„  =  24  inches)  of  the  cylinders  whose  "Axial  Deviation"  is  30°.  On  the 
same  horizontal  ruling  are  given  10.2576  inches  as  the  "Primary  Focus," 
and  149.7422  inches  as  the  "Secondary  Focus."  The  nearest  practical 
equivalents,  expressed  in  refraction,  are  shown  to  be  1-10  and  1-160,  rc- 
spectivety,  which  in  practice  will  be  found  to  be  the  lenses  most  closely  ap- 
proximating neutralization  of  the  principal  meridians  of  the  combined  cylin- 
ders. 

In  the  second  set  of  tables  (2),  under  the  heading  "Elementary  Eefrac- 
tion,"  the  cylinders  are  expressed  in  dioptrics,  and  in  the  right  hand  ver- 
tical column  the  laws  mentioned  on  pages  83  and  84  are  forcibly  exem- 
plified. 

The  Dioptral  Formulas  on  page  82  were  applied  in  these  tables,  and 
will  generally  be  found  most  convenient  for  use  by  the  student  who  may 
desire  to  solve  similar  examples.  In  this  event,  great  care  should  be  exer- 
cised to  retain  the  proper  meaning  and  proportions  of  r^,  r^  and  r^,  as  in- 
dicated by  the  respective  signs  +,  — ,  >  and  <  in  the  left  hand  vertical 
column. 


VERIFICATION    OF   THE   FORMUL-S. 


97 


I.    TABLES  IN  VERIFICATION    OF   THE   DIOPTRIC   FORMUL/E. 

FOR  COMBINED  CONGENERIC  CYLINDERS. 


Elementary 
Foci. 

Axial 
Deviation. 

Primary 
Focus. 

Primary 
Refraction. 

Secondary 
Focus. 

Secondary 
Refraction. 

/,  <  U 

16  3  24 

<i         « 

U                 <l 

y 

t\ 

[Approximale.) 

!<'. 

[Approximale.) 

30° 

45° 
60° 

10.2576 
11.1555 
12.5569 

1/10 

1/11 
1/12 

149.7422 
68  8347 
40.7773 

1/160 

1/72 
1/40 

FOR  COMBINED  CONTRA-GENERIC  CYLINDERS. 


EliEMENTARY 

Foci. 

AxiAIi 

Deviation. 

Positive 
Focus. 

Positive 
Refraction. 

Negative 
Focus. 

Negative 
Refraction. 

/o>fv 

7 

+  i^i 

{Approximafe. ) 

-n 

[Approximate. ) 

—1/32 
—1/22 
—1/16 

—  14C+  10 

tl                      u 

U                           (( 

30° 
45° 
60° 

16.9799 
13  2046 
11.2537 

+  1/16 
+  1/13 
+  1/11 

32.9799 
21.2040 
16.5870 

/o</v 

7 

30° 
45° 
60° 

47.5527 
30.4131 
23.7316 

(Approximale.) 

-Fo 

{Approximate.) 

—1/24 
—1/18 
—1/16 

—  14  C  +  20 
<<                (< 

+  1/48 
+  1/30 
+  1/24 

23.5527 
18.4)31 
15.7315 

2.    TABLES  IN  VERIFICATION  OF  THE  DIOPTRAL   FORMULAE. 

FOR  COMBINED  CONGENERIC  CYLINDERS. 


Elementary 
Refractions. 

Axial 

DevVn. 

Primary 
Refraction. 

Secondary 
Refraction. 

r,>r2 

7 

R, 

(Approx.) 

B, 

(Approx.) 

2.5  C  1.5J9- 

i(                       K 
(I                       « 

30° 
45° 
60° 

3.75D. 

3.46 

3.09 

3.75I>. 

3.5 

3. 

0.25D. 

054 

0.91 

0.25D. 

0.5 
1. 

4D. 
4 

4 

FOR  COMBINED  CONTRA-GENERIC  CYLINDERS. 


Elementary 
Refractions. 

Axial 
Dev't'n. 

Positive 
Refraction. 

Negative 
Refraction. 

Ri  —  Ba= 

ri  —  ro 

ri<—ro 

7 

+  i2i 

(Approx.) 

-B, 

(Approx.) 

+  4C— 2.75Z). 
«            It 

30° 
45° 
60° 

2.397D. 

3.062 

3.564 

+  2  5D. 

+  3. 
+  3.5 

1.147D. 

1.802 

2.314 

— 1.25D. 

—1.75 

—2.25 

+ 1  25Z?. 
+  1.25 
+  1.25 

'•i  <  —^0 

7 

-\-E, 

(Approx.) 

-Bo 

(Approx.) 

'i-  ro 

+  2C— 2.75D. 
<(           « 

30° 
45° 
60° 

0.8561). 

1.325 

1.690 

+  0.75D. 
+  1.25 

+  1.75 

1606D. 

2.075 

2440 

—  1.5D. 

—  2. 

—  2.5 

—  0.75D. 

—  0.75 

—  0.75 

SECTION  III 


THE  PRISM-DIOPTRY 

AND  OTHER 

OPTICAL  PAPERS 

WITH   SIXTY-FIVE  ORIGINAL  DIAGRAMS 


THE  PRISM-DIOPTRY. 


In  the  year  1890  the  author  advocated  a  "  Metric  System  of  Numbering 
and  Measuring  Prisms,"*  involving  the  principle  that  prisms  should  be 
numbered  according  to  their  refractive  powers,  instead  of  by  their  refract- 
ing angles,  or  angles  of  minimum  deviation.  As  prisms  notably  possess 
the  property  of  apparently  changing  the  position  of  objects  seen  through 
them,  it  was  proposed,  in  the  new  system,  that  the  tangent-distance  f 
between  the  object  and  its  virtual  image  should  form  the  basis  of  compari- 
son in  measuring  the  relative  strengths  of  prisms.  The  tangent-deflection 
of  one  centimeter,  measured  in  a  plane  one  meter  from  the  prism,  was, 
therefore,  arbitrarily  though  befittingly  chosen  as  the  new  unit  of  prismatic 
power,  and  was  named  the  prism-dioptry. 

In  measuring  the  refraction  of  prisms,  however,  the  same  as  for  lenses, 
it  is  necessary  that  the  incident  pencils  of  light  should  be  composed  of 
parallel  rays,  so  that  the  theoretical  distance  of  one  meter  must  in  practice 
be  increased  to  at  least  six  meters. 

The  Prismometric  Scale,  |  which  is  to  be  placed  exactly  six  meters  from 
the  prism,  therefore,  represents  the  prism-dioptry  as  a  six-centimeter  dis- 
tance. Scales  which  are  computed  for  a  shorter  distance  than  six  meters 
have  been  placed  upon  the  market,  but,  as  demonstrated  on  page  148,  are 
wholly  unreliable. 

♦Archives  of  Ophthalmology,  Vol.  XIX,  Nos.  1  and  2, 1890 ;  Vol.  XX,  No.  1,  New  York,  1891. 
Archly  fiir  Augenheilkunde,  Band  XXII,  Berlin,  1890. 

The  Ophthalmic  Review,  discus.sion  by  Dr.  Swan  M.  Burnett,  Vol.  X,  No.  3,  London,  1891. 
tOfficinlly  adopted  by  the  section  of  Ophthalmology  of  the  American  Medical  Association,  Waahing- 
ton,  D.  C,  1891.    "To  Mr.  Prentice  alone  belongs  the  credit  of  having  proposed  as  a  standard  prism 
one  which  produces  a  deflection  of  one  centimeter  at  one  meter's  distance,  and  no  advocate  of  the 
centrad  ever  hinted  at  it  until  the  appearance  of  his  paper  in  the  Archives  of  Oi)hthalmology.    We  owe 
the  simplicity  of  that  idea  to  Mr.  Prentice ;  let  us  not  deprive  him  of  whatever  honor  belongs  to  the 
conception."— J/edtcaZ  News,  Philadelphia,  May  2, 1891. 
jThe  American  Journal  of  Ophthalmology,  Vol.  VIII,  No.  10,  St.  Louis,  1891. 
Les  Annales  D'Oculisticuxe,  Paris,  Julv,  1892. 

101 


102 


THE  .PRISM-DIOPTRY. 


The  author  was  the  first  to  recommend  that  the  figure  of  a  prism  /\, 
used  as  an  exponent*  to  the  prism  numerals,  should  be  the  symbolic  sign 
for  the  prism-dioptry,  it  also  being  the  letter  D  of  the  old  Greek  alphabet. 
By  this  means,  one  prism-dioptry  (1^)  is  readily  distinguished  from  the 
prism  of  one  degree  (1°)  refracting  angle,  and,  in  fact,  from  prisms  of  any 
other  system. 

The  Dioptral  f  System  |  of  numbering  prisms  alone  possesses  the  great 
desideratum  of  establishing  a  direct  and  simple  relation  between  the  prism- 
dioptry  and  the  lens-dioptry,  as  demonstrated  by  the  authors'  law,||  that 
' '  a  lens  decentered  one  centimeter  will  produce  as  many  prism-dioptries  as 
the  lens  has  dioptrics  of  refraction."  Thus  a  lens  of  1  D.  decentered  1  cm. 
will  afford  l'^ ;  a  lens  of  2  D.  decentered  1  cm.  Mali  produce  2^,  etc.     The 


Fig.  1. 


Fig.  2. 


prism-dioptral  power  is  also  in  direct  proportion  to  the  amount  of  decentra- 
tion,  so  that  a  lens  of  2  D.  decentered  J  cm.  gives  1^  ;  whereas,  if  the  same 
lens  is  decentered  2  cm.  it  produces  4^,  and  so  on.  It  is,  therefore,  only 
the  size  of  the  lens  which  in  practice  will  set  a  limit  to  its  prismatic  power. 

*  Concerning  this  exponent  (a)  sec  paper  by  Dr.  Swan  M.  Burnett  in  Annals  of  Ophthalmology  and 
otology,  July,  1894,  Transactions  International  Ophthalmological  Congress,  Edinburgh,  1894,  and  the 
Refraetionist,  December,  ]  894. 

The  figure  of  a  triangle,  no  matter  how  placed  in  respect  to  the  position  of  its  sides,  refers  exclusively 
to  the  prism-dioptry,  being  so  recognized  by  American  manufacturers. 

t  First  used  in  Prentice's  Dioptric  Formulse  for  Combined  Cylindrical  Lenses,  monograph,  New  York, 
1888.  "  The  selection  of  this  adjective  would  seem  justifiable,  since  the  unit '  Dioptry '  has  been  chosen 
in  distinction  to  'Diop'ric,'  which,  though  related,  has  another  significance."  Thus,  a  40-inoh  tele- 
scope lens  is  a  member  of  a  dioptric  system,  whereas,  a  1-dioptry  lens  is  specifically  a  member  of  the 
dioptral  system. 

In  the  English  language  we  have  an  analogy  to  dioptry  and  dioptral  in  the  spelling  of  ancestry  and 
ancestral. 

X  "  Having,  by  elaborate  practical  test,  fully  convinced  ourselves  of  the  preeminent  advantages  of 
the  Dioptral  System  in  the  art  of  manufacture,  we  have  discarded  the  old  degree  system  entirely,  and 
are  now  manufacturing  prisms  which  are  more  accurately  ground  than  ever  before."  Circular  issued 
to  the  optical  trade  by  the  American  Optical  Company,  Southbridgc,  Mass.;  also  catalogue,  1894. 

"Our  prisms  are  now  ground  to  conform  to  the  metric  system."  Catalogue  of  the  Bausch  &  Lomb- 
Optical  Company  Rochester,  N.  Y.,  1895. 

I  Text-book  of  Ophthalmology,  page  141,  Drs.  Norris  and  Oliver,  Pliiladclphia,  1893. 
TexMx)ok  of  Diseases  of  the  Eye,  page  201,  Henry  D  Noyes,  M.D.,  New  York,  1895. 


THE   PRISM-DIOPTRY.  103 

In  Fig.  1,  abc  represents  a  vertical  section  of  a  1  D.  plano-convex  lens, 
with  three  parallel  rays  /j,  z,,  Zj,  separated  by  one  centimeter  distances, 
which  are  incident  upon  its  plane  side.  These  rays,  after  refraction,  are 
collectively  directed  to  the  focal  jjoint  v^  and  therefore  suffer  perpendicu- 
lar deflections  in  the  focal  plane,*  dv^  which  are  equal  to  the  correlative 
decentrations  of  the  rays  z\,  z^,  i^  at  their  respective  points  of  refraction. 

As  the  spherical  surface  may  be  considered  as  being  built  up  of  an  un- 
broken succession  of  infinitely  small  prisms  of  gradually  and  slightly  vary- 
ing angles,  it  is  to  be  noted  that  the  three  chosen  prisms,  shown  in  their 
order  of  1^,  2^  and  3^,  correspond  to  the  respective  decentrations  of  1,  2 
and  3  centimeters,  and,  therefore,  produce  correlative  deflections  in  the 
focal  plane,  dv^  exactly  the  same  as  the  spherical  surface  at  the  same 
points  of  refraction.  Some  recent  authors  have  failed  to  comprehend  this 
unequivocal  precision. 

In  Fig.  2  three  concentric  curvatures  are  shown  to  represent,  respec- 
tively, the  spherical  or  cylindrical  surfaces  of  1,  2  and  4  dioptry  lenses,  in 
which  the  same  prism  of  1^  occupies  a  different  position  (decentration), 
relatively  to  the  optical  axis,  on  each  of  the  lenses. 

Beneath  each  section  is  given  the  dioptral  power  of  the  lens,  which, 
being  multiplied  by  the  decentration  in  centimeters,  shows  the  same  pris- 
matic power  of  2^  to  exist  at  a  different  though  definite  point  on  the  sur- 
face of  each  lens. 

Thus  it  is  seen  that  every  lens,  whatever  its  dioptral  power,  contains  all 
possible  values  of  the  prism-dioptry,  which  means  that  the  prism-dioptry 
itself  must  constitute  a  distinct  part  of  every  lens  of  the  dioptral  system. 

The  prism-dioptry,  therefore,  stands  unchallenged  in  its  unique  ability 
to  harmonize  all  of  the  refracting  elements  in  the  optometrical  lens-case 
by  establishing  a  complete  and  inseparable  relationship  between  prisms 
and  lenses.  We  need  only  to  remember  the  centimeter  in  connection  with 
the  prism-dioptry,  as  we  do  the  meter  in  its  relation  to  the  lens-dioptry. 

•"The  great  and  enduring  work  of  Gauss  on  the  elucidation  and  simplification  of  optical  laws  has 
among  its  cardinal  elements  four  planes— the  anterior  and  posterior  focal  planes  and  the  two  principal 
planes  (Haupt-Ebenen);  and  the  proportion  of  the  size  of  image  to  object,  as  elucidated  by  the  formula 
of  Helmholtz,  is  calculated  on  the  tangent  plane.  .  .  .  This  plane  can,  in  the  case  of  prism-deflec- 
tion, be  regarded  in  the  same  light  as  the  focal  plane  of  the  standard  lens.  .  .  .  This  method  was 
first  suggested  and  made  practical  by  Jlr.  C.  F.  Prentice,  of  New  York,  .  .  .  who  has  gone  very 
thoroughly  into  the  mathematics  of  the  subject  in  his  paper."— 77i€  Ophtfialmic  Keview,  a  monthly  record 
of  ophthalmic  science,  London,  England,  January,  1891. 


A  METRIC  SYSTEM  OF  NUMBERING  AND 
MEASURING  PRISMS. 

Revised  reprint  from  the  Archives  of  Ophthalmology,  Vol.  XIX,  No.  1, 1890.    Also  translated  by  the 
author,  and  published  in  Arehiv  f\ir  Anjrenheilkunde,  Band  XXII,  Berlin,  1890. 


Introductory  ier7?ia /•/.>■  by  Dr.  Swan  M.  Burwlt. 

"The  old  method  of  numbering  prisms  simply  by  the  angular  deviation  of  their  sides  is,  confessedly, 
inaccurate  and  unscientific.  Any  attempt  to  .supplant  this  by  one  more  accurate,  and  to  place  the  no- 
menclature of  prisms  on  the  same  basis  of  scieutiflc  exactness  as  the  other  optical  appliances  in  the 
hands  of  the  practical  ophthalmologist  is,  therefore,  deserving  of  consideration.  The  method  proposed 
by  Mr.  Prentice,  in  the  following  paper,  not  only  docs  this,  but  does  it  in  a  manner  and  according  t<> 
principles  which  are  familiar  to  even  the  less  sicentitic  practitioners.  To  have  the  same  unit  (the 
meter)  of  measure  and  comparison  for  all  refracting  apparatus  and  imiform  with  the  nomenclature  em- 
ployed in  the  designation  of  anomalies  of  refraction  and  muscular  equilibrium,  gives  a  simplicity  which 
is  not  only  commenda,ble  in  itself,  but  tends  to  render  the  study  of  the  practical  use  of  prisms  easier  and 
more  comprehensible  to  the  student.  This  is  particularly  apparent  in  the  connection  the  author  estab- 
lishes between  the  prism-dioptry,  the  lens-dioptry  and  the  meter-angle.  Not  the  least  important  part  of 
the  contribution  is  the  description  of  the  instrument  Mr.  Prentice  has  devised  for  illustrating  his  idea 
and  for  testing  the  relraction  of  prisms  generally." 

The  present  method  of  designating  prisms  by  the  angular  deviation  of 
their  refracting  surfaces,  is  open  to  the  objection  that  we  thereby  define 
only  an  isolated  feature  of  their  construction,  to  the  utter  disregard  of  the 
varying  powers  of  refraction,  which  must  result  from  the  use  of  refracting 
substances  having  different  indices  of  refraction. 

With  a  view  to  securing  greater  accuracy  and  uniformity  in  our  utili- 
zation of  the  refractive  properties  of  prisms,  the  following  system  of  num- 
bering, which  the  author  believes  to  be  feasible,  as  well  as  suited  to  the 
requirements  of  optometrical  practice,  is  presented. 

IjQi  abc,  Fig.  1,  represent  a  prism,  with  the  ray  z  incident  perpendicularly 

to  ab,  and  we  shall  have  dv  as  the  deflection  accruing  from  the  refraction 

at  e.    Similarly,  d  V  will  represent  the  deflection  arising  from  the  refraction 

at  the  same  point  e^  for  a  prism,  ABC,  of  greater  angle. 

105 


106         A   METRIC   SYSTEM   OF   NUMBERING   AND  MBASDRING   PEI6MS. 

We  shall  then  have 


But 


dV  :  de  =  d{V^  ;  </,<? 

djV^  =  dv  =  6 

dV  _    3 
de  d,e 


Fig.  1. 


By  adopting  one  meter  as  the  distance  of  the  point  e  from  the  plane  /Vj, 
in  which  the  amplitudes  of  deflection  dv  and  dV  are  measured,  we  shall 
have  </tf  =  1,  when 


dV  = 


d,e 


(1) 


It  is  further  evident  that  the  greater  the  refractive  power  of  the  prism, 
the  greater  will  be  its  correlative  deflection  in  this  plane,  so  that 

R  :  dV=:  r  :  3 (2) 

where  R  and  r  represent  the  refractive  powers  of  the  prisms  ABC  and  abc, 

respectively. 

Consequently,  R  =  dV,  M^hen  r  =  <5. 

Substituting  these  values  in  the  equation  (1),  we  have 

d^e 
or,  if  d^e  =  x„  as  a  function  of  the  meter, 

ADowing  r  to  represent  the  unit  of  prismatic  refraction,  we  then  have 


A   METRIC   SYSTEM    OP   NUMBERING   AND   MEASURING   PRISMS. 


107 


Consequently,  the  prismatic  refraction  is  in  inverse  proportion  to  the  dis- 
tance at  which  the  unit  deflection  is  produced,  being  fully  in  harmony 
with  the  refraction  of  a  lens,  which  is  in  the  inverse  proportion  to  the  dis- 
tance at  which  the  image  is  formed. 

Provided,  therefore,  a  standard  amplitude  of  deflection  be  adopted  as  the 
unit,  and  which  shall  be  measured  in  a  plane  one  meter  from  the  refract- 
ing surface  of  the  prism,  we  shall  be  enabled  to  designate  prisms,  in  spe- 
cific terms  of  dioptrics,  for  instance,  with  the  same  significance  as  in  lenses. 

Thus  prisms  of,  say,  two,  three  or  four  prism-dioptries  will  produce  the 
same  unit-deflection  at  one-half,  one-third,  or  one-quarter  of  a  meter, 
respectively. 

By  the  relation  (2)  we  shall  also  find  the  same  prisms  to  produce  two, 
three  or  four  times  the  unit-deflection  in  the  meter-plane^  so  that  the  prob- 
lem reduces  itself  to  the  selection  of  a  series  of  prisms  which  shall  produce 
tangent-deflections,  at  this  distance,  which  are  multiples  of  the  adopted 
unit. 

As  the  deviation  produced  by  a  prism  will  be  dependent  upon  its  angle 
and  the  index  of  refraction,  we  ma}'  here  note  the  relation  existing  be- 
tween these  factors,  when,  as  in  Fig.  2,  a  ray  i  is  incident  perpendicularly 
to  the  face  ab^  and,  consequently,  suffering  refraction  at  e  only.     Here  we 


, 

< 

A          ^^P 

d 

p. 

'  V^''^^'^'"^---^ 

> 

c 

c 

^ 

Fisf.  2. 


have  <^  abc  =  /5,  as  the  angle  of  the  prism ;  <^  iep^  =  /?,  the  angle  of  inci- 
dence ;  '^pev,  the  angle  of  refraction  ;  <:^  dev  =  r,  the  angle  of  deviation. 


Consequently  the  index  of  refraction 

sin  <^pev sin  (<^  ped  +  <X  ^^'^) ^^^  (/'  +  ?')  .  .   .    (1. ) 


sin  <^  iep^ 


sin  <^  iep^ 


sin  /? 


108         A    METRIC    SYSTEM    OF    NUMBERING    AND    MEASURING    PRISMS. 

siti  Y 
This  equation  may  also  be  given  the  form  :  ta7tg  {i  =  — ^ — .     .    (2.) 

when  it  is  desired  to  determine  the  angle  (/3)  of  the  prism  for  any  known 
value  of  the  angle  (/)  of  deviation. 

As  the  angles  of  prisms  which  we  shall  here  have  to  consider  are  compar- 
atively small,  we  shall  use  the  above  instead  of  the  formula  for  mini- 
mum deviation. 

A  relation,  therefore,  exists  between  rj^  ,3  and  y,  which  requires  that  two 
of  these  factors  be  known  to  enable  us  to  determine  the  third,  so  that  our 
choice  of  a  unit  deflection  will  be  included  in  the  following  propositions, 
wherein  prisms  of  low  degree,  the  usual  limits  of  refractive  index,  and 
comparatively  small  deviations  of  the  refracted  ray  will  have  to  be  con- 
sidered : 

Proposition    I. — The  values  of  ij  and  ,3  being  given  to  find  y 


II.— 

( ( 

u 

"-/ 

(C 

r 

C  I 

u 

iC 

C  I 

;3 

III.— 

1  ( 

( i 

1 

u 

/3 

i  i 

i  i 

11 

(l 

V 

For  purposes  of  illustration  and  reference,  we  may  consider  only  the  first 
proposition,  for  the  generally  accepted  index,  ij  =  1.53  (Spiegelglass),  for 
a  l'^  prism,  when  we  have,  from  (1)  : 

^^^^i^=l-5o.-..s-/;z(l°+r)=  1.53X0. 017452  =  0.026701  =::fi7zl°31'.8 
stn  1  . 

sin  (1°  +  rj  =  sin  1°  31'  48" 
.-.  r  =  31' 48" 
The  deflection  >\  corresponding  to  the  angle  y,  being  equal  to 

cie.  tang  ^  =  1„,  tayig  y  =  fang  31'  48" 
we  have,  in  meters,  '^  =  0.009250. 

A  prism  producing  a  deflection  equal  to  the  tangent  of  31'  48",  equal  to 
0.00925  at  a  distance  of  one  meter,  will  therefore  correspond  to  the  accu- 
rately ground  prism  of  one  degree  refracting  angle,  with  an  index  of  1. 53. 

In  case  glass  of  another  index  were  used,  it  would  be  necessary  to  vary 
the  angle  of  the  prism,  so  as  to  satisfy  the  conditions  of  refraction  for  pro- 


A   METRIC    SYSTEM    OF   NUMBERING    AND   MEASURING    PRISMS. 


109 


ducing  the  aforesaid  deflection,  and  it  is  therefore  obvious  that  manufac- 
turers will  be  privileged  to  adopt  any  correlative  proportion  of  angle  to  in- 
dex which  will  satisfy  the  demands  for  any  tangent-deflection  which  it  may 
be  determined  to  adopt  as  a  unit. 

Supposing  the  chosen  unit  of  deflection,  tang  ^,  to  be  slightly  greater 
than  the  above,  say  exactly  equal  to  0.01,  or  one  centimeter,  our  series  of 
prisms  would  then  be  : 

1  Prism-dioptry  producing  a  tangent  deBection  =  1"»  in  the  meter-plane. 


=  2«»      " 
=  3«"      " 


A  system  of  numbering  prisms  in  terms  of  prism-dioptries,  could  therefore 
be  adopted  which  would  satisfy  all  the  conditions  here  set  forth. 

Such  prisms  could  be  measured  by  noting  the  deflection  they  produce 
upon  the  index  line  of  a  coarse  centimeter  scale,  placed  at  right  angles  to 
the  line  of  sight,  at  the  distance  of  one  meter.*     (See  Fig.  3.) 


y~ 


Fig.  3 


While  a  restriction  of  this  character  offers  the  advantage  of  a  ready 
ocular  means  of  verifying  the  correctness  of  the  prisms,  there  are  at  present 
however,  many  difficulties  to  be  overcome  in  manufacturing  them.  Calcu- 
lation would  disclose  the  fact  that  such  prisms  would  require  to  be  ground 
to  degrees,  minutes  and  seconds,  so  that  comparatively  few  prisms  out  of  a 
lot,  at  the  close  of  our  effort  to  produce  them,  would  be  found  to  actually 
meet  the  aforesaid  requirements. 

This  would  so  heighten  their  cost  as  to  render  them  impracticable, 
except  as  diagnostic  instruments  in  the  consultation  room. 

Even  these,  however,  could  be  substituted  by  the  prism-mobile,  which 
consists  of  two  prisms  rotating  before  each  other  in  opposite  directions,  and 


*  For  reasons  given  in  the  paper  on  the  Prismometric  Scale,  perfect  accuracy  will  only  be  insured 
■.vhcn  this  scale  is  enlarged  to  the  dimensions  required  for  its  use  at  a  distance  of  six  meters. 


110     A    METRIC    SYSTEM    OF    NUMBEKING    AND    MEASURING     PRISMS. 

which  will  afford  the  most  ready  means  of  filling  a  demand  for  definite  de- 
flections, inasmuch  as  the  rotation  of  the  prisms,  from  0°  to  180°,  pro- 
duces all  possihle  deflections  from  one  millimeter  upward. 

The  instrument  could  easily  be  graduated  to  read  to  centimeters,  and 
tenths,  or  millimetei's  of  deflection.  For  the  determination  of  muscular 
insufficiencies  of  comparatively  low  degree,  and  to  render  the  instrument  as 
light  as  possible,  a  special  cell,  to  contain  weak  rotating  prisms,  could  be 
devised,  similar  to  that  of  Dr.  Eisley,  to  fit  the  trial  frame. 

We  may  venture  to  assert  that  the  prism,  although  the  simplest  element 
in  Dioptrics,  is  the  most  difficult  to  manufacture,  when  required  to  be 
exact;  and  we  shall  therefore  be  obliged,  for  the  present  at  least,  to  use 
existing  commercial  prisms  for  spectacle  glasses. 

We  shall  subsequently  show  that  these  may  be  profitably  utilized,  by 
assigning  the  unavoidable  variations  of  deflection,  consequent  to  the  manu- 
facture of  such  prisms,  to  their  proper  places,  as  members  of  the  new 
system. 

By  actual  experiment  the  author  has  found  imported  prisms,  represented 
as  being  of  one  degree  (1°),  to  produce  defiections  varying  between  9  and 
13  millimeters,  and  which,  if  reduced  to  the  basis  of  our  standard  of  one 
centimeter,  are  to  be  designated  as  0.9  and  1.3  prism-dioptries,  respectively. 

Similar  discrepancies  of  deflection  are  found  to  exist  throughout  the 
entire  series  of  imported  prisms  now  in  use,  so  that  we  shall  have  adequate 
variety,  covering  almost  every  required  interval  of  the  new  system;  where- 
as, by  the  earlier  method,  an  optical  prescription,  although  required  to  be 
exact,  is  constantly  exposed  to  the  danger  of  being  reduced  to  little  else 
than  a  ticket  of  chance  in  an  optical  lottery. 

Farther  on  an  instrument  will  be  described  which  the  author  has  devised 
to  determine  the  power  of  prisms  in  dioptries  and  fractions,  thus  making 
it  possible  for  the  optician  to  select  from  his  stock  the  one  that  shall  fill 
the  requirements  of  deflection  sought.  Indeed,  by  its  use,  we  may  hope  to 
have  manufacturers  ultimately  furnish  us  prisms,  in  packages,  assorted  and 
marked  with  the  number  indicating  their  power  in  dioptrics.* 


*•  This  expectation  lias  been   fulfilled  by  American  manufacturers   ever  since  1S94,  see  page  102. 


A    METRIC   SYSTEM   OF   NUMBERINQ    AND    MEASURING    PRISMS. 


Ill 


THE   RELATION    OF   THE    PRISM-DIOPTRY    TO  THE    METER- 
ANGLE. 


In  the  accompanying  Fig.  4,  E^  and  E.^  represent  the  centers  of  rotation 
of  the  eyes,  and  OM  the  median  Hne,  bisecting  the 
base-line  E^  E^  at  M. 

For  the  point  of  fixation  6>,  corresponding  to  the 
angle  of  convergence,  y^  for  the  eye  E^^  we  have 


b  =  ti8 


«.< 


sin  Y^ 


0E\ 


b 


AIM. 


or  if  C,  =  1  meter,  sin  y^  =  b  =  the  deviation  re- 
quired of  the  eye  which  is  optically  adapted  for  the 
point  O. 

The  base-line,  b,  having  a  constant  value  for  each 
inter-pupillary  distance,  23,  we  have  the  angle  y^ 
solely  dependent  upon  the  varying  metrical  values 
of  Cj,  so  that  the  unit-angle  of  convergence  has  been 
designated,  by  Nagel,  the  meter-angle,  when  Q  =  1 
meter.     Hence  1  ma  =  arc  siji  -7^-=  arcsm  -t-=  b ; 


arc  sin 


arc  S171 

b 


Ci 


arc  sin  t7-=2  b; 

7i 


2  ma  =  arc  sin  -^ 
a.  s.  f. 

We  may  now  proceed  to  find  the  value  of  the 
prism  in  prism-dioptries,  which,  being  placed  before 
the  eye,  with  its  base  in,  shall  substitute  an  effort 
of  convergence  to  the  same  point. 

The  deviation  produced  by  a  prism  of  one  prism- 
dioptry,  at  one  meter,  being  equal  to  <5,  Fig.  4,  it  is 
evident  that  the  prism  which  shall  be  required  to 
produce  a  deviation  b  =^  b^=^  ^^,  will  have  to  be  i? 
""     b        '  times  greater  than  one  prism-dioptry  (l^),t  conse- 

^'^•^'  quently  =  ^.l^ 

If  the  same  deviation  is  to  be  produced  at  the  dis- 
tance a,  =  —  of  one  meter,  when  C^  =  1  meter,  for  the  meter-angle,  the 
prism  will  require  to  be  Y  times  as  great,  consequently 


t  See  page  102. 


112         A    METRIC   SYSTEM   OP   NUMBERING   AND   MEASURING   PRISMS. 

1  ma  =  Kry.l^ 
But  V='^   and  7  =  ~ 

Similarly,  if  the  deviation  d  is  to  be  produced  at  the  distance  a^  =z  ~ 
of  one  meter,  when  Q  =  ^  meter,  for  two  meter  angles,  the  prism  will  re- 
quire to  be  Z times  greater  than  tj.I^,  or  Z.-q.l^.     Consequently 

2ma  =  Z.-qA^ 
But  Z  =  —  and  v  =  ~r 
.•.2ma  =  ^.  -^  .  1^ 

o 
.-.  ;^war=  J-  .   4.    1^  =  '-5!^1A    .       .       .       (I) 

When  convergence  is  confined  to  comparatively  small  angles,  we  may 
regard  the  sine,  tangent,  and  angle  as  being  equal  to  one  another.  In 
other  words,  we  may  consider  a^  =  C^  =  1,  a.,  =  Q  =  ^,  etc.,  so  that 

l».a=i.A.l^  =  A.i. 

2  raa  =  -!-  .  4-  .  1'  =  2   '-  .  1' 

"2  0  0 

Under  thes^  circumstances  one  prism-dioptry  differs  from  the  meter- 
angle  by  the  co-efficient  -y-.  This  is  due  to  the  selection  of  a  compara- 
tively small  unit-deflection  5.  If  we  had  chosen  a  greater  unit-deflection, 
say  S  =  d,  then  one  meter-angle  would  correspond  to  one  prism-dioptry 
exactly.  A  prism,  producing  so  great  a  deflection  as  half  the  inter-pupil- 
lary distance,  would,  however,  give  too  great  an  angle  for  the  unit  and 
lowest  degree  of  prism,  unless  others,  as  fractions  of  the  unit,  were  in- 
cluded in  the  series.     For  instance,  if  5  =:  ^  equal  the  deflection  for  one 


A    METRIC    SYSTEM    OF   NUMBERING    AND  MEASURING    PRISMS. 


Ill 


prism-dioptry,  prisms  of  lesser  refraction  might  be  designated  as  0.25"^, 
O.S'^,  0.75^,  when  their  respective  deiiections  are  }  b,  ^  b,  and  f  b. 

Such  a,  selection  would,  however,  possess  no  particular  advantages,  since 
b  will  generally  be  a  variable  quantity  for  different  individuals  ;  besides, 
it  will  not  be  admissible  to  approximate  the  factor  —  for  considerable 
degrees  of  convergence. 

A  strict  consideration  of  the  co-efficient  „ — . ,  under  such  circumstances, 
will  be  imperative,  without  regard  to  any  particular  choice  of  the  unit 
deflection,  so  that  for  considerable  degrees  of  convergence  the  prism-diop- 
tries  will  have  to  be  determined  by  the  Formula  I. 

The  following  table  exhibits  the  errors  committed  in  estimating  the 
value  of  the  angle  of  convergence,  when  Formula  II  is  substituted  for 
Formula  I,  in  reducing  meter-angles  to  prism-dioptries. 


METER-ANGLES  REDUCED  TO  PRISM-DIOPTRIES,  FOR  AN  INTER-PUPILLARY  DIS- 
TANCE OF  64  MILLIMETERS  (3  =  32  "•/„,).     STANDARD  UNIT  DEFLECTION 
FOR  1  PRISM-DIOPTRY  =  '^  =  0.01  =  !""■  AT  THE  METER-PLANE. 


Distance  from  the 
Object  of  Fixa- 
tion to  the 

Sine 
of  the 
Meter- 
Angle. 

Tangent 
of  the 
Meter- 
Angle. 

Value  of  the  Angle  of  Convergence. 

Value  of  the  Angle  of  Convergence 

when  sine,  tangent  and  angle 

are  accepted  as  equal. 

Eye. 

In 

Meter- 
Angles. 

In  Prism- 
Diop- 

tries. 

In  Degrees  ex- 
actly =  A  De- 
viation. 

In 
Meter- 
Angles. 

In  Degrees 
arc  sin. 

In  Prism- 
Diop- 
tries. 

In 
Meters. 

In  Milli- 
meters. 

A  Deviatitm 
arc  tang. 

1 

i 
i 

\ 

20 

1,000 
500 
333.3 
250 
200 
100 
50 

0.032 

0.064 

0.096 

0.128 

0.160 

0.32 

0.64 

0.0320163 

0.0641309 

0.0964465 

0.129061 

0.162080 

0.337765 

0.832919 

1 
2 
3 
4 
5 
10 
20 

3.20163 
6.41309 
9.64465 
12  9061 
16.2080 
33.7755 
83.2919 

1°50'    1"6 
3°  40'    9"  9 
5°  30' 32"  1 
7°21'14"4 
9°  12' 24'' 6 
18°  39'  46"  7 
39° 47' 30" 

1 
2 
3 
4 
5 
10 
20 

1°50'  1"5 
3°  40'  3" 
5°  30'  4"5 
7°  20'  6" 
9°  10'  7"5 
18°  20'15" 
36°  40'30" 

3.2 
6.4 
9.6 

12.8 

16 

32 

64 

1°  49'68"2 
3°  39'43" 
5°  29'  0"9 
7°  17'38"9 
90    5'24"8 
17°  44'40" 
32°  37'  8"5 

It  is  apparent  that  such  a  substitution  will  be  admissible,  up  to  five 
meter-angles,  where  the  difference  between  the  deviation  produced  by  the 
prism,  and  the  value  in  degrees  of  five  meter-angles  amounts  only  to 
4'  42"  7. 

When  we  consider  that  a  muscular  insuflQciency  of  five  meter- angles  is 
entirely  beyond  the  limits  of  optical  correction,  the  latter,  in  fact,  being 
confined  to  deficiencies  of  about  one  meter-  angle,  or  less,  it  is  obvious  that 
a  substitution  of  the  sine  for  the  tangent  will  be  justifiable. 


114         A   METRIC    SYSTEM   OP   NUMBERING   AND   MEASURINa    PRiSMS. 

Such  being  the  case,  the  subject  of  prismatic  corrections  becomes  wonder- 
fully simple.  For  instance,  for  an  inter-pupillary  distance  of  60  millimeters 
=  6  "",  the  base  line  will  be  3  ™,  when,  according  to  Formula  II, 

1  fna  =  j^  =  3  prism-dioptries. 

Similarly,  for  an  inter-pupillary  distance  of  50  millimeters,  the  base-line 
being  25  millimeters,  equal  to  2.5  centimeters,  the  meter-angle  will  be 
equal  to  2.5^. 

Thus,  for  each  inter-pupillary  distance,  we  find  a  different  prism  neces- 
sary to  supplant  the  meter-angle.  This  is  but  natural,  since  greater 
demands  for  convergence  will  be  necessary  in  wide  than  in  narrow  inter- 
pupillary  distances. 

This  leads  us  to  the  final  and  simple  rule  : 

Read  the  patient's  inter-pupillary  distance  in  centimeters, 
when  half  of  it  will  indicate  the  prism-dioptries  required  to 
substitute  one  meter-angle  for  each  eye. 

One  could  scarcely  hope  for  a  more  convenient  method  than  to  find  the 
prism-dioptries,  corresponding  to  one  meter-angle,  expressed  in  the 
patient's  features. 

There  will,  however,  frequently  be  occasion  to  supply  less  than  one 
meter-angle,  as  indicated  in  the  following  tabulated  examples. 


Pupillary  distance,  2Z»  ==  56        60 

64        68        millimeters 

Base-line     ...      6  =    2.8       3 

3.2       3.4     centimeters 

1  meter-angle .     .     .   =    2.8      3 

3.2       3.4     prism-dioptries 

}  meter-angle .     .     .   =    0.93     1 

LOG     1.13 

^  meter-angle .     .     .   ^    1.4       1.5 

1.6       1.7           "         " 

According  to  our  standard,  a  prism  of  0.9'^  will  produce  a  tangent 
deflection  of  0.9  of  a  centimeter,  or  9  millimeters,  and  a  prism  of  1.1'^  a 
deflection  equal  to  1.1  centimeters,  or  11  millimeters.  It  will  therefore  be 
possible  to  select  these,  by  aid  of  an  adequate  instrument,  from  a  paper  of 
I*'  prisms  of  foreign  manufacture,  since  the  latter  are  found  to  produce 
deflections  varying  between  the  same  limits. 

Prisms  of  1.3'^  to  1.6^  will  similarly  be  found  among  prisms  of  1^°, 
and  so  on. 


A    METRIC   SYSTEM    OP   NUMBERING    AND    MEASURING    PRISMS. 


115 


Later  we  shall  describe  the  instrument,  called  a  prismometer,  to  distin- 
guish it  from  optical  theodolites  and  goniometers  of  the  physical  laboratory, 
which,  we  believe,  offers  an  advantage  over  the  present  methods  of  meas- 
uring prisms,  inasmuch  as  it  makes  it  possible  to  measure  prisms  to  the 
nicety  of  fractions.  * 

The  general  principle  evolved,  therefore,  affords  a  new  means  of  verifying 
the  correctness  of  prisms  in  a  simple  manner,  and  must  assuredly  serve  its 
purpose,  whichever  standard  unit-deflection,  at  one  meter  distance,  it  may 
be  ofl&cially  determined  to  adopt,  although  it  is  believed  that  the  centimeter 
has  been  shown  to  possess  such  decided  advantages  as  to  be  worthy  of 
favorable  consideration. 


THE     RELATION     OF    THE    PRISM-DIOPTRY    TO     THE    LENS- 

DIOPTRY. 

Some  of  the  advantages  of  the  prism-dioptry,  as  the  unit  of  measure  for 
the  refraction  of  simple  prisms,  having  been  shown,  and  whereas  prisms 
are  frequently  combined  with  spherical  lenses,  it  is  here  proposed  to 
further  consider  the  relations  of  the  prism-dioptry  to  such  combinations, 
as  well  as  to  the  equivalents  which  are  to  be  obtained  by  a  mere  decentra- 
tion  of  the  spherical  lenses  themselves. 

In  the  accompanying  figure  5,  a  lens  is  shown  with  its  principal  anterior 
and  posterior  foci  /and  F,  equidistant  from  O,  upon  the  optical  axis/OF. 


Fig.  5. 

The  ray,  ze,  which  is  parallel  to  the  optical  axis,  and  incident  at  an 
eccentric  point  e,  of  the  lens,  being  refracted  to  the  focal  point  F  will  sus- 

*  In  the  original  paper,  the  first,  and  consequently  more  or  less  cnide  prismometer,  was  described  as 
being  capable  of  measuring  prisms  up  to  20  a.  As  in  subsequent  papers  this  was  shown  to  have  been  an 
error,  the  reader  is  referred  to  the  descriptions  of  the  Perfected  Prismometer,  and  the  Prismometric  Scale. 


116 


A    METRIC    SYSTEM    OF    NUMBERING    AND   MEASURING    PRISMS. 


tain  a  deflection  dF,  in  the  focal  plane,  which  will  always  be  equal  to  the 
decentration  Oe. 

A  ray,  de^  which  is  parallel  to  the  optical  axis  and  incident  from  the 
opposite  direction,  will  be  refracted  to  the  focal  point/,  and,  if  received  by 
the  eye  at  E^  will  be  projected  by  it  in  the  prolongation  oife  to  v. 

Consequently  dv  =  <y  will  be  the  measure  of  the  apparent  displacement 
of  the  point  d,  resulting  from  the  prismatic  action  inherent  at  e.  By  refer- 
ence to  the  figure,  and  previous  definitions,  we  then  have 

<  (^ev  =  <^  0/e 
<^  de  F  =  -^  O  Fe 
But  <  Ofe  =  ^10  Fe 
.-.    '^dev=<^deF 
d  =  dF=Oe 

The  tangent  deflection  'J,  al  the  focal  plane  of  thelens^  is  therefore  always 
equal  to  the  amount  of  decentration,  and  consequently  in  direct  proportion 
to  it. 

Augmented  decentration  of  the  lens  will  be  associated  with  an  increase 
n  the  prismatic  action,  resulting  from  a  growing  inclination  of  the  tangents 
/j  /,  .  .  .  determining  the  obliquity  of  the  spherical  surfaces,  at  corre- 
sponding opposite  points  of  eccentricity,  as  shown  in  Fig.  6. 


Fig.  6. 


Fig.  7. 


The  inclination  of  these  tangents,  relatively  to  each  other,  becomes  a 
maximum  when  the  angle  between  them  reaches  180°,  that  is  to  say,  when 
they  both  coincide,  and  so  form  the  tangent  to  the  lens,  which  must  then 
have  become  a  perfect  sphere.  The  amount  of  prismatic  action  it  will  be 
possible  to  obtain  consequently  only  depends  upon  the  diameter  of  the  lens. 
It  will  be  well,  however,  to  bear  in  mind  that  lenses  of  large  diameter, 
whose  spherical  aberration  for  peripheral  rays  causes  the  latter  to  fall  short 


A   METRIC    SYSTEM    OF    NUMBERING    ANT)   MEASURING   PRISMS. 


117 


of  the  focus,  will  naturally  also  effect  a  still  further  increase  in  the  pris- 
matic action  at  the  focus,  for  an  extreme  decentration. 

It  is  further  obvious  that  a  virtual  prism  of  constant  angle  cannot  exist 
within  the  lens  to  produce  the  aforesaid  variable  results.  The  Fig.  7  is 
therefore  apt  to  prove  misleading,  as  there  is  but  one  fixed  amount  of 
decentration,  d^  Fig.  8,  which  corresponds  to  a  prism  of  the  angle  /?,  and 
which  is  determined  by  the  tangents  to  the  lens  surfaces  at  «?,  drawn 
parallel  to  the  sides  of  the  inscribed  prism. 

Thus  prepared  we  may  proceed  to  a  consideration  of  the  prism-dioptry. 

Supposing  a  lens  to  be  1  D.,  and  the  deflection  sought  to  be  1™'  =  1-^  at 
the  meter-plane,  which  is  the  focal  plane  of  the  lens,  it  would  require  a 
decentration  of  one  centimeter  to  produce  this  result. 

A  lens  of  2D.,  being  decentered  the  same  amount,  would  produce  a 
tangent  deflection  equal  to  I*'"'  at  its  focus,  half  a  meter^  which  would  be 
equivalent  to  2""  at  the  meter-plane,  or  2-^,  it  having  been  shown  that  the 
prismatic  refraction  is  in  the  inverse  proportion  to  the  distance  at  which 
the  unit-deflection  is  produced.  The  following  tabulation  will  serve  to 
make  this  clear. 


Lens. 

Decentration. 

Tang.  Deflections  at  the 
Focus. 

Tang.  Deflections  at  the 
Meter  Plane. 

ID. 
2D. 

3D. 

1'='"  at  1  meter 
!<='"  at  \  md,er 
l"*"  at  \  meter 

Jem  JA 

2cm  _-  2a 

3«"»  ==  3a 

This  table  also  reveals  the  unique  law  that: 

Any  lens  is  capable  of  producing  as  many  prism-dioptries  as 
the  lens  possesses  dioptries  of  refraction,  provided  it  is  de- 
centered  one  centimeter. 

On  the  other  hand  the  prism-dioptries  will  increase  or  decrease  as  the 
decentration  becomes  greater  or  less.     Thus  : 


Lens. 

Decentration  in 
Centimeters. 

Decentration  in 
Millimeters. 

\  cm. 

2  «"»• 

1"/ 

S""/™ 

e-lr. 

—  ~ 

0.25  D. 
0.5    D. 
0.75  D. 

1  D. 

2  D. 

0.25 

0.5 

0.75 

1 
2 

0.6 

1 

1.5 

2 

4 

— Prism- Dioptries  — 

u                    (< 
((                <( 
l(              (( 

0.025 

0.05 

0.075 

0.1 

0.2 

0.076 
0.15 
0.226 
0.3 
0.6 

0.15 

0.3 

0.45 

0..6 

1.2 

118 


A   METRIC   SYSTEM    OF   NUMBERING    AND  MEASURING    PRISMS. 


A  lens  of  2  D.,  limited  by  its  size  to  a  decentration  of  3'"/,„,  will  aSord 
0.6'^,  whereas  a  lens  of  1  D.,  capable  of  a  decentration  of  G^/n,,  will  pro- 
duce the  same  prismatic  effect,  as  shown  above.  In  other  words,  a  lens 
of  one-half  or  one-third  the  power  will  require  to  be  decentered  twice  or 
three  times  as  much  to  secure  the  same  number  of  prism-dioptries. 

This  we  find  graphically  demonstrated  in  the  accompanying  figures,  in 
which  the  dimensions  of  decentration  and  lens-curvatures  are  exaggerated 
to  better  serve  our  purpose. 


Fig.  9. 


>/0><i' 


Fig.  10. 


In  Fig.  9  the  lens  of  2D.,  with  a  decentration  d^  =  d  "'/«.,  produces  a 
deflection  of  0.6'^  af  the  meter  plane ^  by  reason  of  the  obliquity  of  the 
lens-surfaces,  determined  by  the  tangents  at  e^  constituting  a  virtual  prism 
of  the  angle  /?,  with  its  apex  at  a^ 

In  Fig.  10  the  lens  of  1  D.,  whose  radius  r^  =  2  rj,  is  decentered  by  the 
amount  d.^  =  2  d^=:z  6"*/^,  to  produce  0.6^  at  the  meter  plane  by  a  virtual 
prism  of  exactly  the  same  angle  /5,  with  its  base  at  e.^  e,^^  and  apex  at  a,. 

We  have  consequently  but  to  remember  that : 

The  prism-dioptries  in  decentered  lenses  are  in  direct  propor- 
tion to  their  refraction  and  decentration. 

It  is  therefore  actually  possible  to  determine  the  dioptral  power  of  any 
pair  of  contra-generic  lenses  which  neutralize  each  other,  and  whose 
power  may  be  unknown.  All  that  is  necessary  is  to  place  the  lenses  over 
each  other,  and  to  separate  their  optical  centers  exactly  one  centimeter, 
when  the  prism-dioptral  power,  as  read  from  the  Prismometric  Scale*,  will 
be  equal  to  the  dioptral  power  of  the  lenses. 

*  Bee  method  of  manipulation  on  page  140. 


A   METRIC    SYSTEM    OF   NUMBERING    AND   MEASURING    PRISMS. 


liy 


Our  present  lenses  will,  however,  not  permit  of  a  decentration  of  1"", 
owing  to  their  limited  size,  yet,  if  required,  larger  ones  could  be  furnished 
to  the  trade,  at  a  comparatively  small  increase  in  cost,  which  could  be 
utilized  specifically  in  corrections  involving  prism-dioptries. 

However,  in  cases  where  inadequate  size  of  the  lens  prevents  decentration, 
it  becomes  necessary  to  add  to  it  a  constant  prism.  Since  the  slightest 
decentration  of  a  simple  lens  is  certain  to  produce  a  prismatic  effect,  it  is 
evident  that  the  constant  value  of  any  prism,  upon  one  of  whose  surfaces 
a  spherical  lens  has  been  ground,  Avill  only  be  retained  when  the  optical 
axis  of  the  lens  strictly  coincides  with  the  visual  axis.     See  Fig.  11. 

The  effect  of  decentration  will  naturally  be  to  increase  or  diminish  the 
prismatic  action  of  the  constant  prism,  which  has  been  combined  with  the 
lens. 


() 


Fig.  11. 


Fig.  12. 


Fig.  1.3. 


Thus  in  Fig.  12  the  prism  is  increased  by  the  prismatic  action  due  to 
decentration  of  the  lens,  by  shifting  the  visual  axis  toward  the  apex  of  the 
constant  prism,  while  in  Fig.  13  it  is  decreased  by  a  decentration  in  the 
opposite  direction. 

Supposing  a  5  D.  lens  to  be  combined  with  a  prism  of  2'^,  the  former 
being  decentered  2  ""/„ ,  by  shifting  the  visual  axis  toward  the  apex  of  the 
prism.  We  know  that  a  5  D.  lens  will  produce  5^  when  decentered  1°"", 
and  therefore  will  produce  0.2  of  5^  when  decentered  2  millimeters,  which 
is  equal  to  1^.  The  constant  prism  of  2^  has  therefore  been  increased  by 
1'^,  making  it  3^.  A  decentration  of  the  lens  to  an  equal  amount  in  the 
opposite  direction  will  leave  but  1^  for  the  entire  combination. 

Two  millimeters  have  in  this  case  affected  the  value  of  the  constant 
prism  by  50%  of  its  active  function. 

We  can  now  realize  the  importance  and  necessity  of  an  accurate  adapta- 
tion of  spectacles,  with  regard  to  the  inter-pupillary  distance,  when  high 
spherical  corrections  are  resorted  to. 

The  prism-dioptry  and  the  meter-angle  being  directly  dependent  upon 


120    A    METEIC    SYSTEM    OF    NUMBEEINO    AND    MEASUEING    PEISMS. 

the  inter-pupillary  distance,  it  behooves  us,  in  any  endeavor  to  secure  accu- 
rate results,  to  be  exceedingly  particular  as  to  its  measurement. 

Dr.  Stevens  having  fully  sho^vn  the  disturbances  occasioned  by  hyper- 
phoria, we  may  here  be  permitted  to  call  attention  to  the  danger  of  arti- 
ficially producing  it  by  improperly  centering  the  lenses  in  the  vertical 
meridian  in  a  simple  case  of  hyperopia  in  which  no  real  hyperphoria  exists. 

We  shall  admit  that  the  lenses  are  decentered  vertically,  in  opposite 
directions,  by  5  *"/,«  above  and  below  the  horizontal  plane,  which  will  be 
equivalent  to  a  deccntration  of  one  of  the  lenses  by  1  centimeter,  provided 
the  other  is  properly  centered. 

Our  hyperope  being  corrected  by  lenses  of  +  2  D.,  would,  under  these 
circumstances,  be  forced  to  overcome  3^  at  the  meter-plane,  and  therefore 
a  vertical  diplopia  of  12  centimeters  at  6  meters,  which  amounts  to  20 
meters  at  one  kilometer,  or,  approximately,  in  round  numbers,  66  feet  at  a 
distance  of  3,281  feet. 

This  shows  that  the  vertical  adjustment  of  the  lens-centers  before  the 
eyes  should  receive  fully  as  much  attention  as  their  horizontal  distance 
apart. 

While  emmetropes  may  have  one  eye  much  higher  than  the  other,  and 
still  enjoy  comfortable  binocular  vision,  yet  the  same  discrepancy  in  ocular 
elevation  in  an  ametrope  whose  glasses  have  been  fitted  ^nth  their  optical 
centers  on  the  same  horizontal  plane,  will  produce  great  discomfort  on 
account  of  one  of  the  lenses  projecting  its  image  eccentrically  to  the  macula 
of  at  least  one  eye.  Measurement  by  the  phorometer  will  in  this  instance 
reveal  an  apparent  hyperphoria,  which  in  reality  does  not  exist  as  a  mus- 
cular anomaly. 

Inversely,  in  making  Dr.  Stevens'  test*  for  hyperphoria,  in  emmetropia 
for  instance,  supposing  a  means  to  be  devised  to  enable  the  patient  to 
exactly  indicate  the  distance  between  the  images  which  he  sees  at  a  6-meter 
distance.  Admitting,  by  way  of  illustration,  that  he  has  decided  them  to 
be  6  centimeters  apart,  vertically,  which,  being  equivalent  to  1  ""  at  1  meter 
distance,  will  lead  us  at  once  to  decide  that  a  prism  of  1"^,  properly  placed 
before  the  eye,  will  correct  his  manifest  hyperphoria.  The  same  patient 
would  have  to  struggle  with  a  vertical  diplopia  amounting  to  3^  "*/„i  at  a 


*Punetional  Nervous  Diseases,  by  George  T.  Stevens,  M.D.,  Ph.D.,  New  York,  1887,  page  194. 


A    MKTiaC    SYSTEM    OF    NUMBERING    AND    MEASUUING    rillSMS.      121 

reading  distance  of  33;^  "",  being  quite  sufficient  to  cause  consecutive  lines 
of  small  type  to  appear  intermingled  with  each  other. 

Shortly  after  tiiis  paper  was  first  published,  the  author  devised  the  fol- 
lowing simple  apparatus  (phorometric  chart)  for  estimating  the  deviations 
of  the  visual  axes. 

The  chart  here  described  has,  however,  been  plagiarized  by  one  to  whom 
it  had  previously  been  exhibited. 

In  connection  with  it  the  author  preferably  uses  a  -f-  1^  f)-  cylindrical 
lens,  which  produces  a  much  heaver  line  of  light  than  the  Maddox  rod. 


Fig.  14. 

Fig.  14  shows  a  blackboard  20  inches  square  having  on  its  surface  eight 
vertically  and  eight  horizontally  arranged  dots,  which  are  separated  by  6 
centimeter  distances,  so  that  each  interval  of  space  between  the  dots  repre- 
sents 1"^  at  6  meters  distance  from  the  eye.  A  light  is  placed  behind  a  piece 
of  red  glass  in  a  circular  opening  in  the  center  of  the  board. 

The  patient  being  directed  to  look  through  his  distance  glasses — at  the 
central  light  with  both  eyes,  while  the  cylindrical  lens  is  held  by  the 
operator,  first  vertically  and  then  horizontally  before  one  eye — will  be  able 
to  promptly  indicate  any  deviation  of  the  red  line  of  light  from  the  center, 
by  stating  through  which  of  the  dots  the  red  line  seems  to  pass. 

For  instance,  should  the  red  line  of  light  appear  to  pass  horizontally 
through  the  second  dot  above  the  center,  wlien  the  cylindrical  lens  is 
properly  held  by  the  operator  before  the  patient's  right  eye,  we  at  once 
decide  that  a  prism  of  2^,  placed  with  its  base  up  before  the  patient's  right 
eye,  should  cause  the  red  line  to  drop  two  points  to  the  center,  thus  eorroot- 
ing  the  manifest  vertical  deviation  of  his  visual  axes. 


122         A    METRIC    SYSTEM    OF   NUMBERING    AND   MEASURING    PRISMS. 

Lateral  deviations  of  the  visual  axes  are  to  be  similarly  determined  by- 
means  of  the  horizontal  dots  on  tlie  board.  *  Great  caution  must  be  exer- 
cised, however,  in  reaching  conclusions  respecting  lateral  deviations,  as 
prismatic  corrections  in  such  cases  are  rarely  so  satisfactory  as  when  pre- 
scribed vertically.  It  is  of  course  of  the  utmost  importance  that  the  centers 
of  the  distance  glasses  worn  during  these  tests  should  be  carefully  adjusted 
in  respect  to  their  inter-pupillary  distance  and  elevation. 

As  a  third  demonstration,  supposing  it  to  be  desired  to  afford  binocular 
vision  to  an  emmetrope  at  a  distance  of  J  of  a  meter,  without  his  powers  of 
accommodation  ayid convergence  being  called  into  requisitio7i.  His  inter-pupil- 
lary distance  being  60"V,„,  for  instance,  half  of  it  will  be  3  centimeters,  which 
gives  us  3^  for  his  meter-angle,  making  9'^  requisite  to  set  aside  his  con- 
vergence to  A^  of  a  meter,  while  3  D.  of  lenticular  refraction  are  necessary  to 
substitute  accommodation  for  the  same  distance.  A+3  D.  lens  of  sufficient 
size  would  require  to  be  decentered  3""  to  afford  9^^,  so  that  a  pair  of  such 
lenses,  placed  before  the  eyes  with  their  bases  in,  would  accomplish  the 
desired  binocular  result.  Two  properly  centered  lenses  of +3  D.  combined 
with  prisms  of  9"=^  would  serve  the  same  purpose. 

The  above  illustrations  sufHce  to  show  the  value  of  the  prism-dioptry  in 
leading  to  our  conception  of  the  actual  work  performed  by  prisms  at  dif- 
ferent distances,  and  which  the  degree-system  of  numbering  must  continue 
to  keep  us  in  ignorance  of. 

Besides,  a  degree-system  of  numbering  prisms  cannot  be  brought  to  a 
convenient  relation  to  any  of  the  following  considerations,  which  have  been 
shown  to  exist  in  favor  of  the  metric  system  : 

1.  A  direct  relation  between  the  meter-angle  and  the  prism-dioptry  for 
variable  inter-pupillary  distances,  f 

*  When  the  deviation  of  the  visual  axes  exceeds  the  amount  provided  for  by  the  chart  (4  a),  the  case 
may  be  properly  considered  to  indicate  surgical  intervention. 

t  This  fully  meets  the  suggestion  of  Dr.  Maddox,  who,  in  speaking  of  the  decision  of  the  committee  of 
the  American  Ophthalmological  Society,  consisting  of  Drs.  Edward  Jackssn,  S.  M.  Burnett,  and  Henry 
D.  Noyes,  that  all  ophthalmological  prisms  should  be  marked  with  the  angle  by  which  they  deflect  rays 
of  light,  in  his  work  entitled  "  Ophthalmological  Prisms,"  on  p.  30,  says  :  "  If  I  may  be  allowed  to  sug- 
gest it,  a  still  better  plan  would  be  to  have  all  prisms  marked  with  meter-angles  and  their  fractions,  so 
as  to  correspond  with  lenses  in  the  trial  case,  a  meter-angle  being  the  chosen  unit  of  convergence,  just 
as  a  dioptry  is  that  of  accommodation.  The  only  disadvantage  is  that  the  meter-angle  is  an  inconstant 
quantity." 

This  inconstancy  is  also  mentioned  in  the  first  paragraph  on  page  113  of  this  paper. 

It  has  however,  been  satisfactorily  shown  that  this  seeming  objection  is  an  advantage  to  the  new 
system,  since  the  present  inconstancy  of  our  prisms  can  be  utilized,  thereby  securing  a  degre«  of 
accuracy  unattainable  by  any  other  means. 


A    METRIC    SYSTEM    OP   NUMBERING    AND    MEASURING    PRISMS.         123 

2.  A  direct  relation  between  the  prism-dioptry  and  the  Icns-dioptr}', 
for  any  amount  of  lenticular  decentration. 

3.  Simple  measurement  of  the  inter-pupillary  distance  determining  the 
prism  which  expresses  the  meter-angle. 

4.  All  fractional  intervals  of  the  prism-dioptry  being  rendered  available 
for  differing  inter-pupillary  distances. 

5.  The  prism-dioptry  being  capable  of  measurement  by  a  simple  instru- 
ment, obviating  it  being  taken  for  granted  that  the  prisms  have  been  cor- 
rectly numbered  and  "marked." 

6.  The  resultant  deflection  produced  by  similarly  placed  superposed 
prisms  of  low  power  being  equal  to  their  sum  expressed  in  dioptrics. 

7.  Can  be  applied  to  the  existing  stock  of  prisms,  without  increasing  the 
cost,  or  rejecting  the  present  marketable  product. 

The  latter  is  certainly  a  most  commendable  feature,  since  any  attempt  to 
introduce  prisms  of  special  glass,  or  such  that  produce  definite  angular 
deviations,  will  heighten  the  cost,  while  depreciating  the  commercial  value 
of  those  now  on  hand.  However,  American  manufacturers  have  now 
surmounted  this  obstacle  in  their  production  of  prisms  of  the  dioptral 
system. 

A  possible  objection  to  the  new  system  of  measuring  might  arise  in 
the  fact  that  it  does  not  define  the  minimum  deviation,  yet,  as  the  prisms 
used  in  spectacles  are  of  small  angles,  ' '  the  difference  is  so  trifling  that  it 
may  be  neglected  in  ophthalmic  practice;"*  and,  eo  long  as  it  is  under- 
stood that  prisms  of  greater  angle  are  to  be  held  with  one  side  parallel  with 
the  vertical  inter- pupillary  plane,  from  the  eye,  which  is  the  case  in  meas- 
uring by  the  prismometer  or  prismometric  scale,  we  at  least  obtain  the 
desired  uniformity. 

In  the  event  of  its  being  desired  to  determine  the  resultant  prismatic 
action  of  prisms  which  have  been  combined  at  any  angle  of  crossing,  we 
liave  merely  to  resort  to  the  statical  formula : 


where  P  and  Q  represent  the  prisms,  expressed  in  prism-dioptrics,  and  f 
the  angle  between  their  base-apex  lines. 

* Ophthalmological  Prisms,  by  Ernest  E.  Maddox,  M.B.,  London,  England,  1889.  page  11. 


124 


A  METRIC  SYSTEM   OF  NUMBERING  AND  MEASURING  PRISMS. 


In  ophthalmic  practice  this  is  apt  to  prove  of  possible  value  only  when 
the  prisms  cross  each  other  at  right  angles,  consequently  when  7  =  90**, 
and  therefore, 

R=V  P'  +  Q' 

The  resultant  prismatic  refraction,  R,  will  be  in  the  plane  which  coincides 
with  the  diagonal  of  the  parallelogram  obtained  by  the  forces  P  and  Q. 


2^=P^ 


Fig.  15. 


This  is  graphically  illustrated  in  Fig.  15.  From  a  scale  of  equal  parts 
lay  off  the  values  for  P  and  ^  (2'^  and  3^)  perpendicularly  to  each  other, 
when  the  length  of  R,  measured  by  the  same  scale  of  equal  parts,  will  repre- 
sent the  power  of  the  equivalent  prism  in  prism-dioptries.  The  position  of 
the  base-apex  line  of  the  resultant  prism  R  is  obtained  by  measuring  the 
angle  a  with  a  protractor  whose  center  is  placed  at  B. 


THE  PERFECTED  PRISMOMETER: 

ITS     PRACTICAL    ADVANTAGES,     CONSTRUCTION,    AND 
VARIOUS     APPLICATIONS. 

Kevisiul   reprint   from  the   ""Arohives  of  Ophthalmology."    Vol.    XX,   Nu.   1.   1891. 


In  the  first  numbers  of  these  Archives  of  the  year  1890,  the  author 
described  "A  Metric  System  of  Numbering  and  Measuring  Prisms/'  which 
represented  the  result  of  a  careful  and  extensive  study  of  the  subject,  due 
to  the  suggestion  of  Dr.  S.  M.  Burnett,  who  had  entrusted  him  with  the 
problem  of  searching  for  a  system  which  should  prove  satisfactory  t<> 
ophthalmologists  as  well  as  avoid  conflict  with  the  practical  methods  of 
manufacturing  opticians.  At  the  close  of  the  author's  investigations,  he 
felt  that  this  had  not  only  been  accomplished,  but  that  also  an  instrument 
in  support  of  that  system  had  been  offered  as  a  valuable  assistant  to 
opticians. 

The  author's  familiarity  with  the  routine  of  manufacture  would  not 
allow  him  to  lose  sight  of  the  practical  side,  so  that  this,  being  a  matter  of 
primary  importance  to  opticians,  was  kept  well  in  view  from  the  outset. 
In  advocating  the  metric  system  and  the  use  of  the  prismometer,  we  shall, 
therefore  here  only  do  so  in  so  far  as  they  relate  to  the  interests  of  manu- 
facturing and  dispensing  opticians;  the  advantages  of  the  system  to 
ophthalmic  practice  having  been  previously  set  forth. 

The  author's  argument  in  favor  of  the  metric  system  was,  and  is  based 
upon  the  former  unavoidable  variability  in  the  angles  of  our  prisms,  and 
which  must  result  from  the  foreign  process  of  manufacture.  Although 
this  has  been  indicated  in  the  previous  papers,  we  shall  here  take  the  liberty 
of  quoting  from  a  paper  read  in  connection  with  the  author's  exhibit  of 

125 


12()  THE   PERFECTED   PRISMOMETER. 

the  prismometer  before  the  ?^ew  York  Academy  of  Medicine,  October 
30,  1890 : 

"It  would,  however,  be  exceedingly  difficult  and  correspondingly  expen- 
sive to  manufacture  prisms  producing  only  fixed  intervals  of  deflection.  To 
render  prisms  sufficiently  inexpensive  as  spectacle  glasses  it  is  necessarj' 
that  they  should  be  produced  in  large  quantities  at  one  grinding." 

"The  process  at  present  consists  in  fastening  a  number  of  slabs  of  glass, 
by  means  of  pitch,  or  other  resinous  material,  upon  a  metallic  surface- 
tool.  The  friction  in  grinding  generates  more  or  less  heat,  which  at  times 
is  sufficient  to  soften  the  pitch  and  cause  it  to  yield  beneath  the  slabs. 
Some  slabs  will  shift  more  than  others,  so  that  the  prism-angles  will  vary 
more  or  less  throughout.  Besides,  the  imderlpng  layer  of  pitch  can  never 
be  of  a  uniform  thickness."  Were  it  not  for  these  facts,  competition  alone 
would  undoubtedly  long  ere  this  have  resulted  in  greater  uniformity. 

By  means  of  the  prismometer  the  author  has  found  prisms,  more 
especially  of  low  degree,  to  vary  Ijetween  ten  per  cent,  and  thirty  per  cent, 
of  their  indicated  numbering. 

It  is  obvious  that  if  manufacturers  were  obliged  to  discard  all  those 
prisms  which  varied  from  desired  fixed  intervals  of  prism-angles,  minimum 
deviation,  or  any  other  designated  deflection,  the  price  would  have  to  be 
increased  on  the  perfect  prisms  sufficiently  to  compensate  for  the  cost  of 
those  rejected,  and  which  would  havo  consumed  equally  as  much  material, 
time,  and  labor  to  produce. 

Without  being  confined  to  the  deflections  which  should,  by  calculation, 
correspond  to  the  prism-angles  and  index,  the  author  found,  by  means  of 
the  prismometer,  among  a  series  of  prisms,  of  best  Parisian  manufacture, 
only  the  following  number  to  produce  deflections  which  were  even  alike : 

Three  doz.  prisma  1°  2°  3°  4"  5° 

Number  alike  6  =  1.1      7  =  2      6  =  3.1    6  =  3.7    8  =  4.6  prism  dioptriea 

Balance  varying  between  0.8&  1.6  1.8  &  2.5  2.6&  3.2  3.4  &  3.9  4.4&  4.8       "        " 

These  prisms  were  taken  from  original  packages,  and  may  be  credited 
with  having  been  made  of  the  same  material,  at  the  same  time,  and  upon 
the  same  tools.  Greater  precaution  on  the  part  of  the  manufacturer  could 
not  be  expected. 

To  the  careful  reader  of  the  author's  papers  it  must  have  been  apparent 


THE  PERFECTED  I'RISMOMETER.  127 

that  stress  had  nowhere  been  laid  upon  the  possibility  of  a  variability  in 
the  index,  but,  on  the  contrary,  that  all  his  deductions  were  referred  to 
the  commonly  accepted  index  of  1.53. 

The  privileges,  however,  were  mentioned*  of  which  manufacturers  might 
avail  themselves,  both  in  respect  to  prism-angle  and  index,  in  seeking 
to  provide  prisms  of  the  desired  properties. 

To  any  one  familiar  with  the  use  of  optical  theodolitesf  and  spectrome- 
tersj  it  must  further  be  apparent  that  an  endeavor  to  measure  tlic  minimum 
deviation,  with  prisms  of  small  angles  especially,  is  very  tedious  and  diffi- 
cult. The  apparatus  is  expensive,  requires  a  degree  of  accuracy  in  manipu- 
lation, and  a  knowledge  in  the  reading  of  verniers,  with  which  opticians 
cannot  readily  be  made  familiar.  To  mount  such  prisms  accurately  upon 
the  table  of  the  spectrometer,  and  to  rectify  the  various  adjustments  of  the 
instrument,  are  tiresome  and  slow  operations  which  alone  are  sufficient  to 
condemn  its  daily  use  by  opticians  whose  work  must  necessarily  be  expe- 
ditious. In  the  physical  laboratory,  however,  the  instrument  is  undoubt- 
edly invaluable.  If  the  use  of  an  instrument  is  to  be  abandoned  in  meas- 
uring the  minimum  deviation  suggested  by  Dr.  Jackson,  we  shall  find  that 
manufacturers  will  simply  divide  the  prism-angles  by  two  (2),  for  the  new 
nomenclature,  and  so  give  us  the  old  culprit  disguised  under  a  new  name. 
There  would  be  great  commercial  convenience  to  be  sure,  in  being  able  to 
dispose  of  the  same  prism  under  two  names,  but  no  reform  in  the  inter- 
est of  scientific  exactness  could  be  effected  without  measurement.  Will  it 
be  policy  under  such  circumstances  to  adhere  to  the  minimum  deviation 
merely  for  principle's  sake?  As  the  prismometer  is  intended  to  measure 
the  refraction  of  prisms,  in  terms  of  the  prism-dioptry,  it  may  be  well,  for 
the  benefit  of  those  who  may  have  found  its  simplicity  obscured  by  the 
mathematical  portion  of  the  previous  papers,  again  to  explain  its  principles 
in  more  simple  and  somewhat  different  terms. 

We  know  that  a  lens-dioptry  is  the  unit  of  refraction,  and  corresponds 
to  a  lens  of  one  meter  focus.  Fig.  1. 


*Fage  109. 

t  Lehrbuch  der  Pliysik.,  Prof.  Joh.  Miiiler,  Braunschweig,   1878. 
%  Practical  Physics,  Glazebrook  &  Shaw,  London,  1889. 
Elements  of  Physical  Manipnlation,  Prof.  Ed.  C.  Pickering,  Boston,   1878. 


128 


THE    PERPECTKD    PRISMOMETBR. 


The  prism-dioptry,  since  lenses  are  but  a  fusion  of  prisms  of  varying 
angle,  may  then  also  be  said  to  be  the  linear  deflection  which  the  refracted 
ray  sustains  at  the  focus  of  a  meter-lens,  when  the  incident  ray  impinges 


1  meter  =  100  centimeters. 
A 


IP.  D. 


!   1  D. 


Fig.]. 


upon   a  peripheral   portion  of  the   lens  one  centimeter  from  the  optical 
center  (Fig.  1). 

The  prism-dioptry  therefore  also  represents  the  measure  of  the  angle  of 
deviation  /'„  for  an  eccentricity  or  decentration  of  one  centimeter  (Fig.  1). 


1  meter  =  lOO""- 


JCWI 


f  2  D.  3^  meter. 


Fig.  2. 


A  ray  impinging  upon  the  same  point  of  a  2-dioptry  lens  (Fig.  2)  will 
sustain  the  same  unit-deflection  at  its  focus  ^  meter,  and  will  therefore  find 
the  measure  of  its  angle  of  deviation  y,,  expressed  by  twice  the  deflection 
at  the  meter-plane,  or  2  prism-dioptries.  A  lens  being  decentered  twice  or 
half  as  much  will  produce  twice  or  half  as  many  prism-dioptries  as  the  lens 
possesses  lenticular  dioptrics  of  refraction.*  The  prism-dioptry  is  therefore 
l)ut  a  sequence  to  the  lens-dioptry.  Nothing  can  be  more  simple.  Thus 
the  prism-dioptry  represents  the  proportion  1:100,  which  is  expressive  of  a 
wrade  of  angular  inclination  in  daily  use  by  engineers  and  scientists  the 
world  over.  To  reduce  prism-dioptries  to  degrees  of  angular  deviation,  it 
is  only  necessary  to  divide  the  prism-dioptries  by  100,  when  they  will 

*  Page  ns. 


THE    I'KRFEOTEl)    I'UISMOMETEU.  lii!> 

represent  the  tangents  to  correlative  angles  in  degrees,  which  are  to  be 
readily  found  in  any  table  of  goniometrical  lines.  Since  different  lenses, 
through  varying  decentrations,  will  produce  different  values  of  the  angles 
of  deviation  'Vj  "^2  •  •  •  ,.  how  will  it  be  possible  to  determine  the  value  of 
such  angles  in  degrees,  minutes,  and  seconds?  The  instrument  is  yet  to 
be  invented.  The  prism-dioptry  and  the  prismometer*  solve  the  problem, 
and  in  a  manner  simple  and  rapid  enough  to  any  one  of  ordinary  intelli- 
gence. 

Since  the  therapeutic  value  of  prisms  is  conceded,  and  their  combination 
with  lenses  in  practice  frequent,  the  prismometer  has  been  constructed  with 
due  regard  to  such  combinations,  making  it  possible  by  its  aid  to  utilize 
to  advantage  the  prismatic  action  due  to  decentration  of  the  lens,  for  the 
purpose  of  offsetting  the  error  which  invariably  exists  in  the  associated 
prism,  after  the  combination  has  been  ground.  Would  it  not  then  seem 
unwise  and  even  arbitrary  to  hamper  the  dispensing  optician  in  the 
practical  fulfillment  of  his  work  by  forcing  him  to  a  system  of  degrees 
merely  because  it  harmonizes  with  the  designation  of  a  strabismus  which 
is  incorrigible  by  prisms,  or  with  the  graduations  found  upon  perimeters, 
ophthalmometers,  etc.,  which  have  no  connection  with  prisms  whatever  ? 

The  metric  system  certainly  possesses  the  commendation  of  reducing  all 
the  glasses  of  the  trial  case  to  a  uniform  nomenclature  in  dioptrics.  This 
alone  should  be  considered  a  practical  advantage,  fully  offsetting  the  merits 
of  a  theoretical  minimum  deviation  which  cannot  at  present  be  expeditiously 
verified  by  any  known  means  of  accurate  measurement. 

With  a  view  to  convenience  and  simplicity,  let  us  learn  to  comprehend 
the  power  of  our  prisms  by  their  limits  of  refraction,  shown  by  the  solid 
triangles  in  the  preceding  figures,  when  it  will  become  wonderfully  easy  to 
fit  these  into  meter-angles,  or  for  that  matter  to  any  other  angles  in  space, 
without  necessarily  confounding  prism-dioptries  with  meter-angles,  or 
meter-angles  with  "deviations  of  the  e3'es  in  height,"  as  stated  by  Dr. 
Landolt.  f  The  latter  mistake  could  only  be  the  result  of  a  misconception 
of  the  definition  of  the  prism-dioptry  and  its  relations  to  the  meter-angle. 

In  recommending  the  metric   system   to   the   profession    and    practical 


*  Now  also  the  Prismometric  Scale. 

t  "On  the  Numbering  of  Prismatic  Glasses,"  ArcTiives  of  Ophth.,  XIX,  No.  4,  1890. 


130  THE    rKRFECTKD    PRISMOMETER. 

opticians,  the  author  in  conclusion  begs  to  call  attention  to  its  superior 
advantages,  as  follows : 

1.  From  a  mechanical  point  of  view,  by  taking  the  unavoidable  difficul- 
ties of  manufacture  into  consideration. 

2.  From  a  commercial  and  pecuniary  point  of  view,  by  avoiding  un- 
necessary expense  in  the  production  of  prisms. 

3.  By  the  prismometer,  which  enables  opticians  to  accurately  fdl  the 
demands  of  the  system.* 

Any  system  which  neglects  these  important  considerations  cannot  be 
considered  progressive,  nor  can  it  effect  a  reform  in  the  present  necessarily 
haphazard  endeavors  of  the  dispensing  optician,  with  whom  so  much  of  the 
blame  and  responsibilit}^  must  rest.  Dispensing  opticians  have  always 
been  on  the  alert  to  meet  the  requirements  of  the  profession,  and  will  no 
doubt  gladly  avail  themselves  of  a  system  and  an  instrument  which  will 
enable  them  to  sustain  their  repvitations  as  mechanicians. 

Taking  all  the  facts  into  consideration,  it  suffices  to  say,  that  we  have 
prisms  of  almost  every  imaginable  deflection  on  hand  in  the  market  to-day, 
so  that  it  merely  requires  an  instrument  of  simple  construction,  which  may 
be  used  in  making  the  proper  selection  with  accuracy  and  despatch,  and  this 
is  precisely  what  is  claimed  for  the  prismometer,  which  it  is  the  author's 
purpose  here  to  describe. 

In  the  accompanying  illustration.  Fig.  3,  the  essential  operative  parts  of 
the  instrument  are  shown  to  be  moimted  upon  a  triangular  truss  which 
is  pivoted  by  a  suitable  joint  to  a  pedestal,  so  as  to  permit  of  convenient 
inclination  of  the  whole. 

The  graduated  bar  is  rigidly  supported  near  its  extremities,  upon  the 
truss,  by  two  short  studs  or  pillars,  the  latter  being  slightly  higher  than 
the  radius  of  the  circular  stage,  which  is  supported  at  its  back  by  a  rod, 
fitted,  sliding,  and  acted  upon  by  a  spring  within  the  bar,  so  as  to  auto- 
matically effect  contact  of  the  face  of  the  stage  with  the  knife-edge,  which 
is  also  mounted  upon  the  truss,  between  the  stage  and  the  pinhole  eye- 
piece. 

♦Since  this  paper  was  written,  American  manufacturers  have  so  perfected  the  art  of 
grinding  that  they  are  now  able  to  furnish  prisms  which  are  accurately  numbered  In 
prism-dioptries,  thus  causing  the  prismometric  scale  to  practically  supplant  the  prismo- 
meter as  an   instrument  for   verifying  measurements. 


THE    PERFECTED    PRISMOMETER. 


i;u 


The  divisions  of  the  graduated  bar,  numbered  2,  o,  4,  up  to  10,  are  placed 
^^  h  h  i)  "P  to  tV  of  ^^^6  meter*  length,  counted  from  tlie  knife-edge, 
which  represents  the  zero-end  of  the  scale.  A  vertical  plane,  arranged  to 
slide  upon  the  graduated  bar,  termed  the  index-plate,  is  provided  with  the 
index-line,  marked  zero  (0),  and  two  graduations  at  the  right-hand  upper 


Fig.  3. 


edge,  marked  1  and  2,  which,  being  equal  to  correlative  centimeter  defec- 
tions at  the  meter-plane,  correspond  to  their  equivalents  in  prism-dioptries. 
To  facilitate  subdivision  of  these  graduations  the  index-plate  is  provided 
with  a  transverse  slide  bearing  its  allotted  part  of  the  index-line,  which  ia 
rendered  adjustable  by  a  milled  head  and  micrometer  screw,  the  first  com- 
plete rotation  of  which  will  cause  this  section  of  the  index-line  to  travel 
from  0  to  1,  the  second  complete  rotation  taking  it  from  1  to  2.  The 
milled  head,  being  divided  into  100  parts,  enables  us,  by  its  graduations, 
to  determine  the  position  of  the  index-line  of  the  transverse  slide,  relatively 
to  the  graduations  upon  the  face  of  the  index-plate,  in  lOths  and  lOOtha. 

*  It  has  been  found  convenient  to  construct  the  instrument  to  half  sc«de  throughout. 


132  THE   PERFECTED   PRISMOMIETER. 

Tims,  in  the  accompanying  figure  (4)  we  read  from  the  face  of  the  index- 
plate  "1"  and  from  the  milled  head  y^ths  and  y^-^ths,  or  1.25  for  the  posi- 
tion of  the  index-line  of  the  transverse  slide. 


As  all  readings  of  deflection  must  be  reduced  to  the  meter-plane^  it  will 
be  necessary  to  note  the  position  of  the  index-plate,  which  must  at  all  times 
correspond  to  one  of  the  graduations  of  the  bar.  Consequently,  if  the  above 
reading  is  taken  from  the  index-plate,  when  placed  at  the  figure  "2"  of 
the  bar,  we  shall  have  twice  the  number  of  prism-dioptries  at  the  meter- 
plane,  or  1.25  X  2  =  2.5^ 

For  a  reading  of  "2,"  from  the  index-plate,  when  placed  at  the  gradu- 
ation upon  the  bar  marked  "10,"  we  have  20"^,  which  is  the  maximum 
measuring  capacity  of  the  instrument.  In  other  words,  it  is  merely  neces- 
sary to  multiply  the  readings  of  the  index-plate  by  that  figure  upon  the  bar 
which  defines  the  position  of  the  index-plate  upon  it. 

Before  placing  a  prism  in  position  for  measurement,  it  is  necessary  to 
carefully  determine  its  base-apex  line.  This  is  accomplished  by  such 
slight  rotary  adjustment  of  the  prism  before  the  eye,  until  a  line,  situated 
at  a  convenient  distance,  is  sighted  as  an  unbroken  one,  being  precisely  the 
same  method  which  we  employ  in  determining  the  axes  of  cylinders.  For 
convenience  of  registration,  ink  dots,  in  collimation  with  said  line,  should 
be  applied  to  the  prism.  The  stage  is  provided  with  a  series  of  horizontal 
lines,  engraved  upon  it,  to  facilitate  perfect  adjustment  of  the  base-apex 
line  of  the  prism,  which  is  to  be  introduced  between  the  stage  and  the 
knife-edge,  with  its  apex  to  the  right,  and  gradually  forced  downward 
while  the  ink  dots  pass  successively  from  one  horizontal  line  of  the  stage 
to  the  other,  until  the  upper  edge  of  the  prism  exactly  bisects  the  circular 
opening  in  the  stage.  In  this  position  the  prism  will  exactly  cover  the 
lower  half  of  the  opening,  while  its  lateral  upper  edge  will  be  in  collima- 


THE    PERPECTBD   PRISMOMETER.  183 

don  with  the  lower  edge  of  the  transverse  slide.  On  completion  of  this 
adjustment  it  is  of  the  utmost  importance  that  the  ink  dots  should  coincide 
with  one  of  the  parallel  lines  of  the  stage.  The  observer's  eye  being  placed 
before  the  eye-piece,  will  now  perceive  the  upper  edge  of  the  index-plate, 
and  the  index-line  at  zero  of  the  transverse  slide,  in  their  true  positions, 
whereas  the  lower  portion  of  the  index-plate,  with  its  index-line,  being 
seen  through  the  prism  below,  will  appear  displaced  to  the  right.  The 
position  of  the  observer's  eye  is  now  to  be  carefully  maintained,  while  the 
graduated  milled  head  is  operated  with  the  right  hand,  until  the  index- 
line  of  the  transverse  slide  has  been  shifted  sufficiently  to  the  right  to  make 
contact  with  the  lower  index-line  seen  through  the  prism.  Perfect  coinci- 
dence of  these  lines  is  necessary  for  an  accurate  determination  of  the  de- 
flecting power  of  the  prism  at  any  distance.  It  will  consequently  be  well 
to  previously  remove  any  roughness  of  the  upper  base-apex  edge  of  the 
prism  by  grinding  it  to  a  flat  dull  edge,  and,  to  be  very  precise,  to  take  the 
mean  of  several  readings  while  the  prism  is  in  an  undisturbed  position. 
As  an  example,  we  shall  suppose  the  prism  to  have  been  carefully  adjusted 
in  the  manner  described,  and  that  our  readings  for  three  positions  upon 
the  bar  from  the  index-plate  are  as  follows : 

2(i  Graduation  of  the  bar,  index-reading  ^  1.57  X  2  =  3.14^ 
3d  "  "         "        "  "         =1.05X3  =  3.15^ 

4th  "  "         "        "  "         =0.78X4  =  3.12'^ 

041  A 

Mean:  -^  =  3.13^  + 

This  precaution,  in  the  interest  of  exactness,  may  appear  to  be  unnec- 
essary to  some,  yet  it  is  here  introduced  as  an  exhibit  in  favor  of  the  capa- 
bilities of  the  instrument. 

The  prismometer  is  particularly  valuable  when  it  is  desired  to  measure 
the  inherent  prismatic  action  of  decentered  lenses,  and  their  combinations 
with  prisms. 

In  such  cases  it  will  be  necessarj'^  to  remove  a  peripheral  portion  of  the 
lens  by  grinding  it  to  a  dull  flat  edge,  as  shown  in  the  accompanying  Fig- 
ure 5. 

The  lens  is  then  to  be  placed  upon  the  stage  with  the  flattened  edge  up, 
so  as  to  cover  half  the  stage  opening ;  the  index-line  of  the  transverse  slide 


134 


THE  PERFECTED  PRISMOMETER. 


having  been  previously  adjusted  to  zero  (0).  If,  in  sighting  through  the 
eye-piece,  the  index-line  appears  disjoined,  it  will  only  be  necessary  to 
shift  the  lens  slightly  to  the  right  or  left  to  re-establish  coincidence  of  the 
lines,  when  the  lens  is  said  to  be  centered.     While  in  this  position  ink  dots 


Fig.  5. 


should  be  placed  upon  the  outer  edges  of  the  lens  over  a  centrall}^  situated 
horizontal  line  of  the  stage,  as  shown.  For  this  centered  position  of  the  lens, 
in  sighting  through  the  eye-piece,  we  shall  find  the  index-line  at  zero  (0)  un- 
broken, while  the  lower  half  of  the  index-plate  will  be  enlarged  or  dimin- 
ished according  to  the  character  of  the  lens  employed.  Supposing  the 
lens  be  3  D.  convex,  we  shall  find  the  index-plate  to  present  this  view 
(Fig.  6)  when  it  is  placed  at  the  graduation  marked  "  3  "  upon  the  bar. 

Normal  Plate. 


Fig.  6. — Magnified  Plate. 


The  lower  half  of  the  index-plate  is  provided  with  a  red  line,  indicated 
by  a  dotted  line  in  the  figure,  corresponding  to  a  deflection  of  l'^,  and  which 
appears  proportionately  magnified.  As  it  will  be  inadmissible,  in  our 
readings,  to  place  a  magnified  scale  on  a  par  with  the  normal  scale  of  the 
prismometer,  it  will  be  necessay  to  register  the  magnified  unit  upon  the 
upper  portion  of  the  index-plate,  for  reference  and  comparison  during  de- 
centration  of  the  lens.     To  accomplish  this  we  displace  the  index-line  of 


TUB    PERFECTED    PRISMOMETER. 


136 


the  transverse  slide  until  it  coincides  with  the  red  line  (dotted  line,  Fig  7), 
which,  as  far  as  the  lens  is  concerned,  now  represents  and  takes  the  place  of 
1^  on  the  index-plate. 


_t^_i 


=E^ 


Di 


Fig.  7. 

Now,  by  slowly  shifting  the  lens  to  the  left,  we  shall  observe  the  lines  of 
the  lower  index-plate  to  shift  to  right  (Fig.  8). 


Fig.  8. 

When  the  3  D.  lens  has  been  decentered  one  centimeter,  experiment 
shows  that  the  lower  black  index-line  cuts  the  index-line  of  the  transverse 
slide  above.  Bearing  well  in  mind  that  the  position  of  the  upper  index- 
line  now  represents  "1,"  and  that  our  reading  has  been  taken  for  a  position 
of  the  index-plate  upon  the  bar  at  "3,"  we  have  3^  as  the  result  of  decen- 
tering  a  3  D.  lens  one  centimeter.* 

In  case,  however,  that  the  refraction  of  the  lens  as  well  as  its  decentra- 

tion  have  not  been  previously  determined  it  will  be  necessary  to  note  the 

following  (see  Fig.  6)  : 

normal  plate  1 


It    is     evident    that    the 


magnified   plate 
normal    prism-dioptries   sought   =  -^ 


=    ~=,      so      that      the 


1 
D 


the  magnified  readings,  for 
convex  lenses,  which  will 
be  when  Z>  >  1,  and  the 
the  diminished  readings,  for 
concave  lenses,  which  will 
be  when  Z)  <  1. 


*  Page  117. 


136  THE   PERFECTED    PRISMOMBTER. 

It  is  therefore  only  necessary  to  divide  the  magnified  or  diminished  read- 
ings by  D. 

The  value  of  Z>,  as  we  have  seen,  is  determined  by  first  centering  the  lens. 
It  will  have  a  different  value  for  different  lenses,  and  will  depend  upon  the 
distance  of  the  lens  from  the  index-plate.  In  fact,  D  represents  the  mag- 
nifying or  diminishing  power  of  lenses  for  any  position  of  an  object,  when 
viewed  through  them,  and  which  may  be  placed  within  their  respective 
focal  distances.  For  the  3  D.  convex  lens,  at  the  graduation  upon  the  bar 
marked  "3,"  measurement  by  the  instrument  shows  Z>  to  be  equal  to  1.2, 
Fig.  7.     Suppose  we  decenter  a  -f  3  D.  lens  until  we  obtain  a  reading  say 

of  0.6^,  which  is  of  course  a  magnified  reading,  we  then  have  — ^  =  -^ 

(magnified  reading)  =  0.5  of  the  normal  prism-dioptry  at  the  distance  '  '3," 
or  1.5  normal  prism-dioptries  at  the  meter  plane. 

In  measuring  sphero-prismatic  lenses  we  shall  therefore  find  that  the 
value  of  the  constant  prism  can  either  be  increased  or  diminished  by  a  de- 
centration  of  the  lenticular  element  of  the  combination,  a  decentration  of 
5°/ni  in  the  above  instance  being  sufficient  to  contribute  1.5'^  ad-  or  ab-duc- 
tive  as  occasion  may  demand. 

By  such  means  it  will  be  possible  to  counteract  the  inaccuracies  which 
invariably  exist  in  the  associated  prism  after  the  combination  has  been 
ground.  When  the  lens  is  combined  with  a  prism,  the  flattened  dull  edge 
should  be  cut  parallel  with  the  true  base-apex  line,  the  latter  being  regis- 
tered with  ink  dots  and  adjusted  upon  the  stage  as  usual. 

The  most  ready  means  of  measuring  such  a  combination — for  example, 
-f-  3  D.  spherical  combined  with  2^  (constant  prism) — will  be  to  place  the 
index-plate  at  the  distance  upon  the  bar  marked  "3,"  when,  as  before, 
the  lens  magnification  Z>  =  1.2,  and  which  may  be  more  conveniently  de- 
termined by  previously  centering  a  spherical  lens  of  the  same  refraction. 
Now,  by  deductive  reasoning,  we  know  that  2  normal  prism-dioptries  will 
be  equal  to  f'^  at  \  the  distance,  and  this  would  require  to  be  1.2  greater 
at  the  same  distance  to  appear  as  the  properly  proportioned  magnified  de- 
flection seen  through  the  lens,  consequently  f  of  1.2  =0.8  magnified  prism- 
dioptries.  We  therefore  set  the  line  of  the  transverse  slide  so  as  to  read 
0.8'^  at  the  distance  marked  "3,"  upon  the  bar,  and  proceed  to  decenter 
the  lens  until  the  lower  index-line  cuts  it,  when  we  shall  have  the  desired 


THE    PERFECTED    PRISMOMETER. 


137 


2  normal  prism-dioptries.  We  may  utilize  the  rule  to  prove  the  result : 
0.8  mag.  prism-dioptries  X  3  =  -jc-  ^o  ^^  ^  n<7r/«a/ prism-dioptries. 
Since  lenses  are  capable  of  providing  as  many  prism-dioptries  as  they  pos- 
sess lens-dioptries  of  refraction,  it  also  follows  that  we  shall  occasionally  be 
enabled  to  secure  a  considerable  proportion  of  prismatic  action  by  decen- 
tration  alone,  provided  the  spherical  lens  is  of  proportionately  greater 
strength.  For  instance,  the  3  D.  lens  will  produce  3'^  for  a  decentration  of 
1""-  so  that  an  available  decentration  of  3^"y,„  could  in  itself  be  relied  upon 
to  furnish  l'^  of  the  2^  in  the  lens  forming  the  subject  of  our  example. 

To  facilitate  measurement  of  concave  sphero-prismatic  lenses,  the  stage 
is  provided  with  a  rotating  disk,  within,  containing  three  prisms  of  varying 
power,  with  their  bases  down,  and  which  may  be  successively  carried 
before  the  lower  half  of  the  opening  in  the  stage  as  occasion  may  demand. 

The  object  of  these  prisms  is  to  counteract  the  prismatic  action  in  the 
yertical  plane,  which  would  otherwise  manifest  itself  by  a  confusion  of  the 
transverse  slide  in  its  contact  with  the  loMcr  portion  of  the  index-plate 


(Fig.  9),  as  a  result  of  sighting  through  the  upper  peripheral  edge  of  a  concave 
lens  (acting  as  a  prism  with  its  base  up)  when  placed  in  proper  position  on 
the  stage.  The  extent  of  the  confusion  of  the  parts,  as  shown  in  the  figure, 
will  naturally  depend  upon  the  strength  of  the  lens,  so  that  rotation  of  the 
disk  will  reveal  the  prism  best  calculated  to  re-establish  contact,  as  shown 
in  Fig.  10. 

Our  choice  of  the  prism  being  made,  the  lens  is  to  be  removed  from  the 
stage  so  as  to  rectify  the  position  of  the  disk-prism  before  the  index-line  at 
aero  (0),  which  should  naturally  present  a  perfect  vertical  line  to  view. 

As  an  example,  let  us  suppose  the  combination  —  3D.  sph.  3  2"^ 
(constant  prism)  to  be  presented  for  measurement.  We  should  first  select  a 
concave  3  D.  lens,  centering  it  upon  the  stage  as  described,  and  discover  a 
confusion  of  the  index-plate,  at  "  8  "  upon  the  bar,  as  shown  in  Fig.  9. 


138 


TUB   PERFECTED   PRISMOMETER. 


It  will  be  found  that  the  first  prism  of  the  disk  proves  sufficient  to 
re-establish  contact,  as  in  Fig.  10, 


I'  I" 


e 


I 

Fig.  10. 

Removing  the  lens,  rectifying  the  disk-prism,  and  replacing  the  lens,  we 
find  the  diminishing  power  of  the  lens  Z?=  0.83.  Our  object  being  to 
secure  2^  at  the  meter-plane,  it  follows  that  \  of  this  will  have  to  be  th(^ 

=  0.67  in  the 


2^ 


reading  from  the  index-plate  at  *'  3  "  upon  the  bar,  or    „ 

absence  of  diminishing  power,  and  consequently  0.67  X  0.83  =  0.55  as  u 

result  of  diminution  by  the  lens. 

The  index-line  of  the  transverse  side  is  therefore  to  be  set  to  0.55.  The 
spherical  lens  is  now  to  be  replaced  by  the  sphero- prismatic  lens,  with  it-; 
base- apex  line  marked  and  adjusted  upon  one  of  the  horizontals  of  the 
stage,  and  shifted  upon  this  to  the  right  or  left,  until  the  lower  index-line 
cuts  the  index-line  of  the  transverse  slide.  While  the  sphero-prismatic 
lens  is  in  this  position,  an  ink  dot  is  to  be  placed  upon  it  at  the  knife-edge, 
as  the  dot  is  intended  to  ultimately  occupy  the  center  of  the  frame  in 
which  the  lens  is  to  be  mounted. 

Such  can  be  the  accuracy  of  the  optician's  work,  with  the  aid  of  the 
prismometer  for  the  metric  system,  and  of  which  oculists  in  America  may 
readily  avail  themselves  by  a  simple  request  to  have  their  diagnostic  prisms 
re-numbered  by  measurement  upon  the  instrument.  By  these  explanations 
the  author  hopes  to  have  succeeded  in  conveying  the  fact,  that  his  object 
has  not  only  been  to  promulgate  a  theory^  but  also  to  render  it  useful  and 
fully  subservient  to  practice^  and  in  the  absence  of  which  it  should,  like 
many  another,  only  live  in  minds,  and  mould  in  books. 


THE  PRISMOMETRIC  SCALE 

Revised  reprint  from  the  "American  Journal  of  Ophthalmology,"  October,  1891. 


During  the  past  two  years  "The  Metric  System  of  Numbering  and 
Measuring  Prisms"*  has  been  a  subject  of  considerable  discussion,  although 
the  exact  nature  of  its  unit,  the  prism-dioptry,  does  not  seem  to  have  been 
generally  understood,  while  its  practical  advantages  to  opticians,  "of  whom 
accurate  work  is  expected,"  have  been  wholly  disregarded  in  some  recent 
criticisms,  in  which  it  has  been  compared  with  Dr.  Jackson's  and  Dr. 
Dennet's  equally  as  scientific  though  less  convenient  systems.  It  is,  there- 
fore, now  proposed  to  call  attention  to  a  still  more  simple  feature  of  the 
metric  system,  with  further  explanations,  yet  with  the  understanding  that 
the  reader  is  familiar  with  its  general  principle  and  applications  aa 
originally  explained. 

The  prismometric  scale,  preferably  drawn  upon  heavy  paper  or  card 
board,  consists  of  a  series  of  numbered  gradations,  " 6  centimeters  apart," 


O 


I 


6 


8        9       JO 


Index 


MM  I  M  I  i  I  I  I  Mil  I  I  III 


Line 


Fig.  1. 


with  an  index-line  at  zero,  longer  than  the  rest,  as  shown  in  Fig.  1,  which 
being  just  six    times  greater   than  the    "coarse  centimeter  scale"    le- 
asee page  105. 

139 


146  THE    PKISMOMETEIC    SCALE. 

ferred  to  in  the  author's  first  paper,  is  intended  to  be  placed  at  a  six 
times  greater  distance,  or  "6  meters"  from  the  eye;  when  simple  prisms 
may  be  measured  by  it  according  to  the  manner  originally  set  forth. 

The  scale  is  also  subdivided  to  quarters,  thus  making  possible  the 
measurement  of  prisms  from  0.85  to  10  prism-dioptries. 

The  average  deflections  produced  by  our  foreign  commercial  prisms, 
marked  1°  to  5°,  will  be  found  to  correspond  closely  to  this  scale  up  to 
the  fifth  division. 

In  applying  the  scale  to  the  measurement  of  sphero-prismatic  lenses,  it 
is  evident  that  the  index-line  will  be  rendered  more  or  less  indistinct  in 
viewing  it  through  such  a  lens,  so  that  the  lenticular  element  of  the  sphero- 
prismatic lens  will  require  to  be  fully  neutralized  by  a  contra-generic  lens 
of  the  same  power,  when,  by  shifting  the  neutralizing  lens  from  right  to 
left,  it  will  be  possible  to  secure  a  position  for  it  which  will  leave  us  the 
prismatic  deflection  which  it  is  sought  to  attain  by  the  inherent  prism  of 
the  entire  combination. 

The  procedure  is  best  explained  by  the  following  example :  The  optician 
"Ijeing  requested  to  grind  a  sphero-prismatic  lens  of  +  3  D.  sph.  3  3^, 
selects  from  his  stock  a  prism  which  is  rough  on  one  side,  and  which  he 
consequently  is  obliged,  from  its  marhing,  to  take  for  granted  is  a  prism  of 
3°.  He  then  grinds  the  rough  side  to  +  3  D.  spherical,  when  according 
to  the  old  method,  he  naturally  assumes  that  he  has  accomplished  the  full 
object  of  his  purpose.  It  is  now  suggested  that  lie  carefulty  determine  the 
optical  center  of  a  concave  lens  of  3  dioptrics,  and  mark  this  point  with  an 
ink  dot,  placing  the  opposite  side  of  this  neutralizing  lens  in  contact  with 
the  spherical  side  of  the  sphero-prismatic  lens  which  it  is  desired  to  meas- 
ure. He  is  next  requested  to  hold  the  entire  combination  before  his  eye, 
at  exactly  6  meters  from  the  scale,  the  precaution  being  taken  to  have  the 
base-apex  line  of  the  sphero-prismatic  lens  horizontal,  with  the  base  to  the 
left,  and  in  such  a  manner  that  the  upper  edge  of  the  entire  combination 
covers  only  the  lower  half  of  the  pupil.  The  index-line  observed  through 
the  lenses  will  then  appear  to  be  displaced  toward  the  right,  relatively  to 
the  graduations  as  seen  through  the  uncovered  upper  portion  of  the  pupil; 
In  the  event  of  the  index-line  appearing  to  be  displaced  more  or  less  than 
the  required  graduation  marked  "2,"  tlio  operator  has  only  to  shift  the 


THE    PRISMOMETRIC    SCALE.  141 

neutralizing  lens  carefully  to  the  left  or  right,  until  the  index-line  exactly 
cuts  the  second  graduation.  Care  should  be  exercised  not  to  change  the 
position  of  the  sphero-prismatic  lens  during  this  act,  and  while  in  this 
position,  an  ink  dot  should  be  placed  on  the  sphero-prismatic  lens,  precisely 
opposite  to  the  dot  on  the  neutralizing  lens.  The  former  then  indicates 
the  point  which  should  form  the  center  of  the  glass  in  the  spectacle  frame. 
The  reasons  for  this  will  be  obvious  from  a  consideration  of  the  following 
figures  : 

A 

Fig.  2.  Fig.  3. 

The  concave  lens  ABC  in  Fig.  2,  with  its  center  at  B,  neutralizes  the 
plano-convex  lens  abc,  thus  securing  the  effect  of  a  prism  acd,  just  at  the 
opposite  points  Bb.  By  shifting  the  neutralizing  lens,  as  shown  in  Fig.  3, 
the  effect  of  a  prism  of  greater  angle  is  obtained.  It  is,  consequently, 
possible,  within  reasonable  limits,  by  this  means  to  correct  any  inaccuracy 
which  may  have  existed  in  the  original  rough  prism.  The  same  effect  is 
obtained  in  sphero-cylindro-prismatic  lenses,  by  neutralizing  the  cylindrical 
element  with  an  additional  and  carefully  adjusted  coiitra-generic  cylindrical 
lens,  though  this  is  naturally  a  little  more  difficult.  Opticians  who  keep 
sphero-cylindrical  lenses  in  stock  will  generally  find  it  more  convenient  to 
use  these  in  neutralizing  compound  lenses  involving  prismatic  power.  It  is 
obvious  that  it  will  be  much  easier  to  hold  and  shift  a  neutralizing  lens 
which  consists  of  only  one  piece  of  glass.  In  shifting  the  neutralizing  lens, 
great  care  must  be  exercised  to  keep  both  cylindrical  axes  parallel  in  case 
a  change  from  their  coincidence  becomes  necessary  to  secure  the  desired 
prismatic  power. 

We  shall  preface  a  further  discussion  of  this  question  with  a  few  simple 
optical  definitions,  which  the  author  holds  to  be  indispensable  to  a  thor- 
ough understanding  of  the  subject,  and  which,  much  to  the  author's  regret, 
and  for  reasons  too  obvious  to  mention,  Avere  not  presented  by  him  in  the 
previous  papers. 

1.  The  optical  center  of  a  lens  is  a  point  situated  upon  a  line  called  the 


142 


THE   PRISMOMETRIC    SCALE. 


optical  axis,  which  must  be  perpendicular  to  both  the  anterior  and  posterior 
surfaus  of  the  lens. 

2.  Direct  Pencils. — Rays  of  light  which  are  emitted  from  a  luminous 
point  upon  the  optical  axis  will  be  refracted  and  directed  to  a  conjugate 
point  upon  the  same  axis,  it  being  specifically  noted  that  the  axes  of  the 
incident  and  refracted  pencils  of  light  and  the  optical  axis  of  the  lens  must 
coincide. 

3.  Oblique  Pencils. — In  any  case  where  the  axis  of  the  incident  cone 
of  light  does  not  coincide  with  the  normals  to  the  surfaces  of  the  refracting 
medium,  whether  it  be  a  lens,  prism  or  plate,  the  refracted  pencil  will  no 
longer  be  a  circular  cone  of  light ;  but,  it  will  be  a  pencil  bounded  by  a 
surface  which  penetrates  and  defines  the  illuminated  area  of  the  medium 
and  two  focal  lines,  which  are  at  right  angles  to  each  other  and  the  axis  of 
the  refracted  pencil  (see  Fig.  6), 


Fig.  6. 


The  same  laws  apply  to  the  reflection  by  spherical  surfaces  of  direct  and 
oblique  incident  pencils  of  light,  and  their  mathematical  elucidation  is 
given  by  Profs.  R.  S.  Heath  and  W.  Steadman  Aldis,  in  their  recent 
exhaustive  treatises  on  Geometrical  Optics. 

In  illustration  of  the  above  definitions,  let  the  curved  line  in  Fig.  4 
represent  the  spherical  surface  of  a  medium  with  a  greater  density  than  air, 


THE   PRISMOMETRIC    SCALE.  143 

when  perpendicularly  incident  conical  pencils  of  light,  projected  upon  it 
from  successive  points  A,  B,  C,  will  have  their  respective  conjugate  foci,  /, 
upon  the  correlative  radii  with  which  the  axes  of  the  incident  pencils 
coincide.  If  the  refracted  pencils,  within  the  medium,  are  to  have  focal 
points  outside  of  the  medium,  the  axes  of  these  pencils  will  have  to  be 
perpendiadarly  intercepted  by  the  second  surfaces  as  shown  by  the  heavy 
lines  in  Fig.  5 ;  and  in  the  event  of  the  second  surface  occupying  an 
oblique  position,  ab^  Fig.  6,  with  respect  to  the  pencil  A^  the  medium 
must  be  considered  as  a  lens,  having  its  optical  center  upon  the  axis  An 
of  the  incident  pencil,  with  the  prism  abc  added  to  it. 

The  circular  cone  of  light,  within  the  medium,  will  then  project  an 
el)ipti(ja,l  area  of  illuminatioti,  Zf,  Fig.  7,  upon  the  second  surface,  as  the 


Fig.  7. 


axis  of  the  pencil  is  here  oblique^  and  the  refracted  pencil  ceases  to  be  a 
circular  cone,  projecting  itself  outside  of  the  medium  as  an  astigmatic 
pencil,  of  which/,  and/,  are  the  focal  lines  at  right  angles  to  the  axis,  the 
whole  being  deflected  toward  the  base  of  the  inherent  prism  P. 


144 


THE    PRISMOMETRIC    SCALE. 


While  this  optical  phenomenon,  which  in  this  case  we  may  term  a 
sphcro-cylindro- prismatic  action,  may  be  new  to  many,  it  has  been  known 
to  physical  science  since  Kummer,  in  1860,  first  called  attention  to  the 
theory  by  which  it  was  mathematically  demonstrated.  Its  significance  to 
optometrical  practice  may,  perhaps,  be  treated  of  in  the  future.* 

The  fact,  however,  may  be  experimentally,  though  crudely,  demon- 
strated by  placing  a  plano-convex  lens  of  8  D.  directly  between  a  light  at  20 
feet,  and  a  screen  receiving  its  image.  On  interposing  a  prism  of  20^,  for 
example,  with  its  base  down,  and  in  a  manner  to  insure  contact  of  the 
plane  faces  of  the  glasses,  the  image  will  be  observed  to  change  both  its 
form  and  position  upon  the  screen.  By  drawing  the  screen  slightly  nearer 
to  the  lens,  a  horizontal  though  imperfectly  defined  line  corresponding  to/, 
will  become  manifest,  and  by  increasing  the  distance  between  lens  and 
screen  a  vertically  elongated  looped  figure,  closely  resembling  a  line  at  f.^, 
will  appear. 

When  a  circular  cone  of  light,  C,  Fig.  8,  from  a  short  definite  distance 
falls  obliquely  upon  the  face  of  a  simple  prism,  we  again  have  an  elliptical 
area  of  illumination,  and  the  refracted  rays  witlibi  the  medium  will 
assume  a  direction  as  if  emitted  from  the  focal  lines  z;„  z\^  reaching  the 
second  surface  of  the  prism,  and  being  refracted  by  it  to  the  eye  at  £",  as  if 
projected  from  the  lines  Fj ,  V.^ ,  on  the  opposite  side  of  the  prism. 

There  is  one  exception  to  this  result,  and  that  is  when  the  axis  of  the 
incident  pencil  assumes  a  direction  which  is  subject  to  minimum  deviation, 


Normal. 


Fig.  8. 

in  which  event  the  emergent  pencil  will  appear  to  diverge  from  a  pointy  at 
the  same  distance  from  the  anterior  surface  as  the  original  source  of  light 
C.     In  the  case  of  a  plate,  the  emergent  pencil  will  also  be  of  astigmatic 

*  Now  mentioned  in  Hand  Book  of  Optics,  for  Students  of  Ophthalmologj-,  W.  W.  Suter,  M.D.,  1899. 


TilE    PRISMOMETRIC     SCALE. 


145 


form,  with  the  difference  that  it  will  appear  to  })rocced  from  a  pair  of  focal 
lines  located  upon  an  \xyj\s.  parallel  to  the  axis  of  the  incident  pencil. 

This  sphero-cylindro-prismatic  action,  on  the  part  of  a  simple  prism 
may  be  experimentally  demonstrated  in  the  following  manner.  Construct 
the  figure  MO  (to  the  left  in  Fig.  9,  in  which  the  width  of  the  principal 
lines  is,  say,  2  inches,  and  the  distance  apart  of  the  perpendiculars  is 


Fig.  9. 


about  24  inches),  and  place  it  at  right  angles  to  the  line  of  sight,  at  a  dis- 
tance of  about  6  feet  from  the  eye,  before  which  a  prism  of  10^  is  given 
considerable  inclination  to  the  visual  axis,  with  its  base  in  or  out,  and  as 
shown  in  the  diagrams,  to  the  right  in  Fig.  9.  The  eye  in  each  instance  is 
to  be  placed  directly  opposite  to  the  figure  M.  In  both  cases  the  prism  is 
shown  not  only  to  have  changed  the  position  of  the  solid  cross  O,  but  also 
to  have  altered  the  dimensions  of  its  vertical  and  horizontal  bars  in  com- 
parison with  M.  "With  these  facts  in  mind  we  may  return  to  our  subject 
of  measurement. 

In  Fig.  10  the  relative  positions  of  the  object  of  fixation  (9,  the  prism, 
and  the  eye  are  shown.  It  is  evident  that  the  perpendicularly  incident 
axis  OV  oi  the  conical  pencil  of  rays  emitted  by  the  object  O  coincides 
with  the  visual  axis,  and  that  the  axis  of  the  refracted  pencil  F/*  does  not 
enter  the  eye,  although  it  does  define  the  deflection  01  which  it  is  desired 
to  ascertain.  The  axis  of  the  refracted  pencil,  d^Ey  which  does  enter  the 
eye,  however,  will  result  from  that  incident  pencil  whose  axis  is  oblique 
relatively  to  the  normal  at  d,  and  it  will  therefore  be  a  ray  approaching 


146 


THE  PRISMOMETRIC    SCALE. 


direction  for  minimum  deviation  and  will  consequently  suffer  less  deflec- 
tion, Oif  than  the  refracted  pencil  whose  axis  is  VP. 

Now  if,  as  is  the  case  with  the  prismometer,  the  observer  reads  the 
deflection  0<  at  the  definite  distance  marked,  say,  "10,"  upon  the  graduated 


0 n 


bar,  it  is  evident  that  an  error  will  be  committed,  since  10  times  Oi  will  be 
less  than  10  times  01;  *  yet  this  seeming  weakness  in  the  author's  pre- 
vious papers  has  escaped  detection  by  the  critics  of  the  prism- dioptral 
system,  and  for  the  consolation  of  whom  let  it  now  be  said  that  there 
could  have  been  no  reasoning  so  clever  or  ingenious  on  their  part  as  to 
have  made  this  error  any  the  less  apparent,  even  in  a  prism  of  io°y  by 
merely  contrasting  the  differences  between  arcs,  sines  and  tangents,  in 
a  choice  for  the  unit  of  measurement.  Besides,  a  mere  consideration  of 
the  well-known  relative  goniometrical  values  of  these  has  not  hitherto 
been  pertinent  to  the  discussion,  since  the  proposed  unit,  the  prism- 
dioptry,  is  not  a  goniometrical  unit,  but  an  optical  unit.  The  desire  to 
multiply  any  unit  in  optics  should  be  curbed  by  a  knowledge  of  the  fact 
that  all  the  fundamental  optical  laws  are  based  upon  the  assumption  and 
acceptance  of  values  of  limited  m.agnitude,  and  that  there  is  therefore  apt  to 
be  a  point  where  unreasonable  multiplication  of  an  optical  unit  will 
contradict  the  actually  existing  optical  phenomenon.     A  warning  to  this 

•This  will  be  equally  true  for  mcasuremants  taken  from  an  arc  at  short  finite  dJatance. 


THE   PRISMOMBTRIG    SCALE. 


147 


effect  was  given  in  speaking  of  the  decentration  of  lenses  (see  page  116  of 
the  author's  second  paper). 

Even  thickness,  a  dimension  which  we  are  taught  to  neglect  with  respect 
to  ophthalmic  lenses,  becomes  an  appreciable  factor  in  prisms  above  8^, 
when  we  attempt  to  measure  their  deflection  at  short  finite  distance.  This 
will  be  apparent  from  the  following  considerations. 

It  has  been  shown  that  the  ray,  which  in  the  nearest  limit  reaches  the 
eye,  is  the  axis  Od,  Fig.  11,  of  an  oblique  pencil,  being  refracted  within 
the  prism  ABC  from  d  to  ^„  and  thence  in  air  to  the  eye  E,  which  projects 
it  to  2,  upon  the  scale  01.  For  a  given  thickness  of  prism,  this  is  the  only 
pencil  which  will  be  received  by  the  eye,  since,  if  we  increase  the  thickness 
by  allowing  the  plane  A^B^  to  represent  the  anterior  surface  of  the  prism, 
the  original  incident  axis  Od  will  be  refracted  at  v  instead  of  </,  when  the 
axis  of  the  refracted  pencil  will  traverse  the  path  vv^P^  to  the  left  of  the 
eye  and  parallel  to  dd^E.  The  refracted  pencil  which  would  enter  the 
eye,  for  the  indicated  increased  thickness,  could  only  accrue  from  an  in- 
creased obliquity  of  the  incident  axis  Ou.  The  latter  would  therefore  even 
more  closely  approach  position  for  minimum  deviation,  from  which  we 
are  to  conclude  that  the  deflection  noted  upon  the  scale  01  by  the  eye  will 
be  least  near  the  base  and  consequently  greatest  near  the  apex  of  the  prism, 

0  il 


4. 

/f 

1   / 

■'l 

r 

Fig.  11. 


This  is  really  proven  to  be  the  case  by  experiment  with  the  prismo- 
meter.  At  the  distance  marked  "10"  upon  the  bar,  the  index-plate- reading 
near  the  base  of  a  22°  prism,   1^  inches  square,  is  found  to  be  1.79, 


148  THE    PRISMOMETRIC    SCALE. 

whereas  at  the  feather-edge  apex  it  is  1.89,  so  that  the  prism  in  the 
former  instance  measures  17.9'^,  while  in  the  latter  it  is  18.9^.  The  same 
prism  measured  by  the  prismometric  scale  at  6  meters  reads  20^.  The 
error  committed  b}^  measurement  througli  ilie  apex,  at  short  finite  dis- 
tance, is  therefore  l.!"^,  while  the  increased  thickness  at  the  base  still  further 
increases  the  error  by  l'^.  The  error  will  consequently  be  least  in  prisms 
of  high  degree,  when  readings  on  the  prismometer  are  taken  at  the  apex  of 
the  prism,  and  it  will  be  reduced  to  a  minimum,  throughout  the  principal 
refracting  plane,  when  the  deflection  is  measured  for  pencils  which  are 
perpendicularly  incident  to  all  points  of  the  prism-surface,  that  is  to  say, 
when  the  pencils  of  light  are  cylindrical,  and  which  will  practically  be  the 
case  when  the  object  of  fixation,  a  line,  is  situated  at  6  meters  distance- 
In  fact  it  will  be  ])etter  to  measure  all  prisms  above  8^  at  this  distance. 

This  sharply  defines  both  Dr.  Burnett's  and  the  author's  reason  for 
advocating  the  tangent  plane  for  the  position  of  the  scale,  since  it  will  be 
infinitely  more  convenient  to  place  such  a  scale  upon  a  flat  wall,  with 
which  every  ofike  and  workshop  is  provided,  than  to  rontrive  an  arc  of  6 
meters  radius. 

Other  advantages  of  the  scale  at  a  6  meter  distance  were  mentioned  in 
the  author's  second  paper,  when  referring  to  hyperphoria. 

The  above  facts  do  not  lessen  the  value  of  the  prismometer,  which  the 
author  has  repeatedly  and  specifically  represented  as  being  of  importance 
to  opticians  in  filling  oculists'  prescriptions,  in  which  the  prisms  do  not 
exceed  5^,  and  by  reason  of  which  the  error  is  so  slight  as  to  be  inappreci- 
able, yea,  even  in  a  prism  as  high  as  8'*',  when  an  attempt  is  made  to 
verify  measurement  by  the  prismometric  scale  at  6  meters. 

It  was  also  to  be  supposed  that  all  oculists  and  opticians  would  not 
provide  themselves  with  prismometers,  in  which  event  it  was  further 
anticipated  that  the  prismometric  scale  would  have  to  be  resorted  to,  and 
more  particularly  now  tliat  hair-splitting  fractions  of  the  unit  are  not  con- 
sidered to  be  of  value. 

A  more  simple  and  convenient  means  of  verifying  the  opticians'  work 
could  certainly  not  be  placed  in  the  hands  of  the  oculist. 

The  prism-dioptry  does  not  exclusively  depend  upon  trigonometrical 
laws,  nor  rest  solely  upon  the  adoption  of  a  specific  instrument,  but  it  is 


4 
TIIK    FRISMOMETRIC    SCALE.  149 

based  upon  a  principle  which  is  easily  understood  and  capable  of  being 
practically  applied  within  the  confining  limits  set  by  the  fundamental 
laws  of  optical  science.  Tt  must  also  be  apparent  that  the  generally  irrel- 
evant criticisms  which  have  appeared  in  print  have  not,  thus  far,  proven 
anything  to  the  contrary ;  while  it  must  be  equally  clear  that  this  paper 
contains  a  review  of  the  optical  laws  and  phenomena  which  must  be  con- 
sidered in  the  choice  of  a  unit,  and  that  these  will  require  to  be  thoroughly 
understood,  before  anyone  can  undertake  a  rational  criticism  of  the  sub- 
ject. We  can,  therefore,  only  admit  that  a  perpetuance  of  the  old  degree 
system,  together  with  the  commonly  accepted  approximations  which  must 
accompany  its  application  in  practice,  will  serve  no  better  purpose  than  to 
■obviate  such  intelligent  pains  being  taken. 


ON  THE  PRACTICAL  EXECUTION  OF 

OPHTHALMIC  PRESCRIPTIONS 

INVOLVING   PRISMS. 

Revised   reprint  from   the   American   Journal   of  Ophthalmology,    January,    1895. 


It  is  intended  here  to  point  out  some  instances  in  which  lenticular  de- 
centration  may  be  taken  advantage  of  in  the  execution  of  prescriptions  in- 
volving a  combination  of  prisms  with  spherical  and  cylindrical  lenses,  in 
a  manner  to  insure  absolute  accuracy,  with  least  inconvenience  of  meas- 
urement, and  with  minimum  expense  of  production.  Within  the  past  few 
years  the  practical  value  of  the  prism -dioptry,  as  a  unit  of  prismatic 
power,  has  been  appreciated  to  such  an  extent  that  the  American  Optical 
Company  of  Southbridge,  Mass.,  and  the  Bausch  &  Lomb  Optical  Com- 
pany, of  Rochester,  N,  Y.,  have  entirely  discarded  the  old  degree  system 
of  numbering  prisms,  having  now  supplanted  it  by  the  prism-dioptry  for 
their  entire  product.  The  prism-dioptry  is  therefore  no  longer  a  subject 
for  scientific  discussion,  but  one  for  practical  consideration,  having  been 
indorsed  in  its  underlying  principle  by  both  the  American  Opthalmolog- 
ical  Society  and  American  Medical  Association,  and  by  two  of  the  most 
progressive  and  largest  manufacturing  establishments  in  the  world.*  It 
may  be  of  interest  to  note  that  the  gross  productions  of  these  firms 
amounted  to  over  $2,000,000  in  1892.  Such  practical  support  certainly 
portends  an  enduring  future  for  the  prism-dioptry,  about  which  so  much 
pro  and  con  has  been  written  since  its  first  appearance  in  opthalmic  litera- 
ture. In  this  paper  it  will  be  taken  for  granted  that  the  reader  is  at  least 
familiar  with  the  principle  of  the  prism-dioptry,  so  that  only  the  relation 
which  exists  between  it  and  the  lens-dioptry  will  here  be  repeated,  to  wit: 


'See  foot  notes,  pages  101  and  102. 

151 


152  OPHTHALMIC    PRESCEIPTIONS    INVOLVING    PRISMS. 

A  lens  which  is  decentered  one  centimeter  will  produce  as  many  priBm- 
dioptries  as  the  lens  has  dioptries  of  refraction. 

A  knowledge  of  this  law  will  frequently  enable  the  optical  practitioner 
to  change  the  form  of  his  prescription  so  that  it  may  safely  be  entrusted 
for  execution  to  any  optician  capable  of  locating  the  optical  center  of  a 
lens,  and  who  may  be  provided  with  no  other  instrument  of  measure  than 
a  pocket  centimeter  scale. 

As  the  dispensing  optician,  generally  speaking,  is  not  allowed  to  exer- 
cise his  own  judgment  in  transforming  prescriptions,  it  is  necessary  that 
the  oculist's  instructions  should  be  explicit  respecting  the  proper  method 
of  putting  his  prescription  into  the  best  practical  form.  A  few  examples 
will  serve  to  illustrate  the  method  of  applying  the  law  of  decentration. 


Prescription  No.  1. 


0.  D.   -f  ;i  J),  sph.  Z   1^  base  out. 
0.  S.    +  ;i  D.  sph.    C   l'^  ^J^'^^t?  out. 


The  usual  practice  is  to  grind  -|-  3  D.  sph.  upon  a  prism  of  1^,  so  that  a 
plano-convex  spherical  element  of  3  D.  is  substituted  for  the  bi-convex  lens 
used  in  the  trial  frame.  In  another  paper,  "The  Advantages  of  the  Sphero- 
Toric  Lens,"  attention  is  called  to  the  positive  disadvantages  of  this  pro- 
cedure, especially  where  high  degrees  of  curvature  are  concerned. 

However,  in  the  above  example  the  ordinary  method  of  grinding  a  spheri- 
cal surface  upon  one  of  the  faces  of  a  constant  prism  is  also  objectionable 
on  account  of  the  increased  cost,  so  that  for  two  very  important  reasons  it 
is  always  preferable  to  resort  to  decentration  of  tlie  lenses,  where  that  is 
possible,  than  to  grind  sphero-prismatic  combinations.  The  aforesaid  pre- 
scription shows  that  the  prism-dioptries  required  are  few  compared  to  the 
lens-dioptries,  so  that  the  law  of  decentration  becomes  available.  In  ac- 
cordance with  this  law,  3  D.  sph.  decentratcd  1  cm.  gives  3'^,  and  since 
only  1^  is  needed,  a  decentration  of  J  cm.  for  each  lens  will  satisfy  the 
requirements.  To  avoid  unnecessary  expense,  the  prescription  should 
therefore  be  written : 


OPHTHALMIC   PRESCRIPTIONS   INVOLVING   PRISMS. 


155 


O.  U.  +  3  D.  sph.  decentercd  ^  cm.  toward  the  temples,  by  which  we 
mean  that  the  thick  edge  of  each  lens  should  be  placed  at  the  temples,  as 
in  Fig.  1,  which  is  the  right  eye. 


The  optician's  ability  to  apply  the  law  of  decentration  is  hampered  only  by 
one  obstacle  at  present,  namely  ^  by  the  limited  size  of  the  lenses  now  furnished 
by  the  manufacturers. 

These  lenses,  as  they  are  usually  supplied  to  the  trade,  are  capable  of  a 
decentration  of  only  J  cm.  laterally,  and  ^  cm.  vertically.  But  even  un- 
der these  circumstances  there  are  many  prescriptions  in  which  decentra- 
tion may  be  profitably  utilized.     For  instance : 


Prescription  No.  2. 


0.  D.  —  2  D.  cyl.  180  ; 
0.  S.  —  2  D.  cyl.  180. 


2^  base  up. 


Might  be  written : 

O.  D.  —  2  D.  cyl.  180  decentered  ^  cm.  down. 
0.  S.  —  2D.  cyl.  180  decentered  ^  cm.  up. 


The  decentration  of  ^  cm.  on  a  2  D.  cylinder  produces  1^,  so  that 
1"^  base  up  on  the  right  eye,  and  1^  base  down  on  the  left  is  equivalent 
to  2'^  base  up  on  the  right  alone.  The  term  "decentration"  signifies  a 
displacement  of  the  lens-center,  relatively  to  the  visual  axis  ;  hence  a  de- 
centration of  ^  cm.  "down,"  on  the  part  of  the  axis  of  the  concave  cylin- 


154  OPHTHALMIC    PRESCRIPTIONS   INVOLVING    PRISMS*. 

der,  means  that  the  axis  is  displaced  downward,  thereby  placing  the  thick 
edge  of  the  lens  up,  as  in  Fig.  2.  The  decentration  would  of  course  have 
to  be  in  the  opposite  direction  for  a  convex  cylinder  to  produce  prismatic 
refraction  with  the  base  up.  Whenever  prismatic  corrections  in  the  verti- 
cal direction  are  necessary,  a  preference  should  be  given  to  place  the  base  up 
before  one  eye  rather  than  base  down  before  the  other,  especially  where  the 
prism  is  stronger  than  2^.  This  is  explained  in  the  fact  that  the  eyes  are 
much  more  frequently  turned  downward  than  upward,  and  are  therefore 
more  exposed  to  the  annoying  internal  reflections,  which  are  noticeable 
near  the  base  of  the  prism,  when  its  base  is  also  down. 


This,  in  part,  also  explains  why  prisms  with  their  bases  towards  the 
nose  frequently  fail  to  prove  so  satisfactory  as  they  otherwise  might. 

As  another  example  let  us  cite  a  case  in  which  the  examination  results  in 

Prescription  No.  3. 

0.  D.  +  4  D.  sph.  O  2^  base  out. 

O.  S.  +  2.75  D.  sph.  O  +  1-25  D.  cyl.  180  Q  2^  base  up. 

In  this  instance  the  lens  of  the  right  eye  would  require  to  be  decentered 
^  cm.  *'out"  to  secure  2^,  since  4  D.  decentered  1  cm.  gives  4^.  We 
know,  however,  that  ^  cm.  decentration  is  in  excess  of  the  amount  (J  cm. ) 
generally  available  in  a  lateral  direction. 

But,  if  we  turn  our  attention  to  the  left  lens,  we  see  that  it  is  possible  to 
call  upon  the  2.75  D.  spherical  to  supply  a  part  of  the  lateral  prismatic  ac- 
tion, as  the  necessary  displacement  can  be  made  along  the  cylinder's  axis 


OPHTHALMIC    PRESCRIPTIONS   INVOLVING    PRISMS.  156 

without  implicating  the  cyhnder.  In  other  words  there  are  6.75  I),  of 
spherical  power  before  both  eyes  which  may  be  called  into  requisition  to 
provide  an  equivalent  to  2^  base  out  for  one  eye. 

The  6.75  D.,  decentered  1  cm.,  gives  6.75^,  or  decentered  one  millime- 
ter =  JLy  of  6.75  =  0.675^  As  2^  are  needed,  it  will  take  3  X  0.675 
=  2.025"^,  that  is  to  say  it  will  take  a  decentration  of  3  millimeters  on  the 
part  of  each  lens  toward  the  temples  to  satisfy  our  requirements. 

The  left  eye  calls  for  2^  base  up.  We  have  vertically  2.75  D.  sph.  -j- 
1.25  D.  cyl.  180,  equivalent  to  4  D.  of  refraction,  which  is  capable  of  pro- 
ducing 2  '^  by  a  vertical  decentration  of  ^  cm.  =  5  millemeters.  The  pre- 
scription should  therefore  read  : 

O.  D.  +  4  D.  sph.  decentered  3  millimeters  out. 

O.  S.  +  2.75  D.  sph.  C  +  1-25  D.  180  decentered  3  millimeters  out  and 
5  millimeters  up. 

These  examples  suffice  to  show  how  easily  and  with  what  absolute  accu- 
racy these  prescriptions  may  be  executed  without  incurring  the  additional 
expense  of  grinding  prismatic  combinations.  This  expense  should  only  be 
incurred  in  those  cases  where  decentration  is  impossible  on  account  of  an 
insufficient  size  of  the  lenses.  A  glance  at  the  prescription  will  determine 
at  once  which  of  the  methods  to  apply.     Take,  for  examble,  a  case  like 


Prescription  No.  4. 

O.  D.  +  1  D.  sph.  O  1^  base  out. 
O.  S.  -f  1  D.  cyl.  90°  O  1^  base  out. 

In  this  case,  as  our  commercial  lenses  are  too  small  to  bear  a  decentration 
of  1  cm. ,  it  would  be  necessary  to  grind  the  lenses  as  the  prescription  ii: 
written,  though  even  here,  to  lessen  the  expense  of  the  left  lens,  it  is  pre- 
ferable to  write : 

O.  D.  4-  1  D.  sph.  C  2^  base  out. 
O.   S.  +  1  D.  cyl.  90° 


156  opirTirALMic  prescriptions  involving  prisms. 

In  this  prescription  care  should  he  taken  to  match  the  lenses  as  nearly 
as  possible  in  thickness. 

It  would  be  a  great  convenience  to  have  the  optical  manufacturers  fur- 
nish a  series  of  lenses  capable  of  a  decentration  of  at  least  1  cm.  Such 
lenses  would  not  require  to  be  larger  than  6  cm.  in  diameter,  and  could  be 
confined  to  the  weaker  powers,  say  from  0.25  to  2  dioptries.  The  cost  of 
such  lenses  should  certainly  not  be  greater  than  that  of  sphero-prisms,  and 
would  offer  many  opportunities  of  applying  the  prism-dioptry  expedi- 
tiously, with  greater  accuracy,  and  less  inconvenience  to  the  dispensing 
optician. 


A   PROBLEM  IN  CEMENTED   BI-FOCAL 

LENSES  SOLVED   BY  THE 

PRISM-DIOPTRY. 

Revise<l  Reprint  from  Annals  of  Ophthalmology  and  Otology,  January,  1895. 


It  is  intended  here  to  illustrate  the  principal  defect  which  so  frequently 
leads  to  disappointment  in  the  use  of  cemented  bi-focal  lenses,  as  well  as 
to  explain  the  technical  means  by  which  it  may  be  prevented.  When  oc- 
casion demands,  it  is  common  practice  among  oculists  to  prescribe  glasses 
for  '  'reading' '  and  *  'distance, ' '  with  rather  vague  instructions  to  the  op- 
tician to  provide  the  necessary  lenticular  corrections  in  the  form  of  bi-focal 
lenses.  These,  in  the  event  of  their  being  of  the  so-called  "cemented" 
variety,  the  optician  executes  by  cementing  wafers  of  glass  to  the  surfaces  of 
the  lenses  which  fit  the  areas  enclosed  by  the  eye-wire  ;  both  of  the  wafers 
being  cut  from  the  peripheral  edges  of  that  lens  which  produces  the  requi- 
site amplifying  or  reducing  power  in  the  lenticular  combination.  The 
principal  effort  of  the  optician,  at  present,  is  to  make  this  lens  as  thin  as 


Fig.  1. 


Fig.  2. 


possible,  and  to  reduce  its  diameter  so  as  to  enable  him  to  secure  at  least 
two  wafers  of  sufficient  size  for  the  ocular  fields  required. 


167 


158 


CEMENTED  BI-FOCAL  LENSES  SOLVED  BY  THE  PRISM-DIOPTRY. 


Economy  and  extreme  thinness  of  wafer  are  doubtless  desirable,  but 
these  are  only  of  minor  importance.  Spectacles  as  now  constructed,  exclu- 
sively with  this  in  view,  are  rarely  ever  free  from  a  prismatic  action,  oper- 
ating vertically,  which  renders  them  very  uncomfortable  to  wear,  and  fre- 
quently useless,  or  harmful.  This  is  especially  true  for  high  degrees  of 
curvature,  and  hi  cases  involving  combination  with  cylindej's,  where  the 
spherical  refraction  is  obtained  by  spherical  curvature  of  07ie  surface  only. 
With  a  view  to  brevit}'^,  only  the  latter  type  of  correction  will  be  discussed. 

The  accompanying  diagrams.  Fig.  1  and  Fig.  2,  will  serve  to  illustrate 
the  defect  referred  to. 

In  each  of  these  the  line  Z,  to  Lis  drawn  upon  the  paper  which  is  sup- 
posed to  be  placed  several  inches  behmd  the  bi-focal  sphero-cylindrical  lens. 
The  line  viewed  through  the  cemented  lens  appears  disconnected,  being 
deflected  by  the  prismatic  action  resulting  from  decentration  of  the  '  'dis- 
tance" and  "reading"  lenses,  relatively  to  each  other.  It  is,  of  course, 
customary  to  have  the  lower  smaller  field  for  reading,  as  above  shown,  but, 
for  convenience  of  easier  demonstration,  the  reader  may  make  the  interest- 
ing experiment  of  superposing  two  concave  lenses,  say  —  4.5  D.  and  — 
2.5  D. ,  on  which  the  optical  centers  have  been  previously  marked  with  ink 
dots,  and  allowing  them  to  occupy  the  positions  shown  in  Fig.  3,  in  which 
the  overlapping  parts  are  in  the  smaller  field  for  distance. 


Fig.  3. 


Fig.  4. 


By  more  widely  separating  the  lens  centers  (4-),  it  will  be  observed  that 
the  disconnected  portion  of  the  line  ZZ,  as  seen  through  the  lenses,  will 
appear  displaced  to  a  lesser  degree,  and,  if  the  upper  concave  lens  2.5  D. 


CEMENTED  BI-FOCAL  LENSES  SOLVED  BY  THE  PRISM-DIOPTRY.  159 

is  chosen  of  sufficient  diameter,  it  will  be  possible  to  secure  a  distance  be- 
tween the  lens-centers  which  will  exhibit  the  line  LL^  unbroken,  as  in  Fig. 
4.  This  shows  that  the  prismatic  action  depends  only  upon  the  distance 
separating  the  lens-centers,  and  therefore  also  that  the  lens,  specifically  in 
this  case  the  upper  one,  from  which  the  segmental  wafers  should  be  cut,  must 
have  a  definite  diameter  for  every  combinatio7i,  if  the  prismatic  action  is  to 
be  eliminated.  We  will  cite  an  example  in  which  we  have  for  distance : 
—  7  D.  sph.  —  2D.  cyl.  ax.  180,  and  for  reading  :  —  4. 5  D.  sph.  —  2D.  cyl. 
ax.  180,  which,  executed  as  a  cemented  bi-focal  lens,  calls  for  a  -j-  2.5  D., 
periscopic  wafer.     This  leads  us  to  the  proposition  : 

What  peripheral  segment  of  a  2.5  D.  periscopic  convex  lens 
should  be  used  as  a  wafer  to  insure  freedom  from  prismatic 
action  in  the  center  of  the  wafer  7  milhmeters  below  the  center 
of  the  distance  lens :  —  7  D.  sph.  —  2  D.  cyl.  ax.  180  ? 

The  key  to  its  solution  is  to  be  found  in  the  law  that  ' '  a  lens  decentered 
one  centimeter  will  produce  as  many  prism-dioptries  as  the  lens  has  diop- 
tries  of  refraction. ' '  * 

The  center  of  the  wafer  being  7  millimeters  (0.7  cm.)  below  the  center 
of  the  distance  lens,  makes  it  obvious  that  we  have  a  prismatic  action  at 
this  point  acting  vertically,  on  account  of  the  —  7  D.  sph.  and  —  2D. 
cyl.  ax.  180,  which  is  equivalent  to  a  decentration  of  0.7  cm.  on  9  D.  to  be 
neutralized  by  the  wafer.  Reverting  to  the  law  we  find  that  9  D.  decen- 
tered 1  cm.  affords  9^,  therefore,  0.7  cm.  will  give  0.7  of  9,  or  6.3^  as  the 
prismatic  action  to  be  overcome. 

The  wafer  of  -f  2.5  D.  decentrated  1  cm,  gives  only  2.5"^,  so  that  it 
takes  a  decentration  of  2.5  cm.  to  produce  2.5  X  2.5^  =  6.25'^.  There- 
fore, this  wafer  when  placed  with  its  thin  edge  at  the  lower  edge  of  the 
concave  distance  lens  will  neutralize  the  existing  6.3'^  Avith  an  error  of 
only  0.05^.  As  will  be  later  shown  the  -f  2.5  D.  lens,  so  as  to  be  large 
enough  for  so  great  a  decentration,  must  be  at  least  64  millimeters  in  di- 
ameter, and  should  be  ground  to  a  knife-edge  to  insure  maximum  thinness. 
As  this  example  came  to  the  author's  notice,  the  instructions  given  to  the 

•  See  page  117. 


160 


CEMENTED  BI-FOCAL  LENSES  SOLVED  BY  THB  PRISM-DIOPTRY. 


mechanic,  who  successfully   executed   the   lenses,    are  here  repeated  as 
follows : 

"Make  a  2.5  D.  periscope  convex  lens  (+  7  D.  —  4.5  D.,)  64  millime- 
ters in  diameter,  worked  to  a  knife-edge  at  the  periphery,  and,  after  mark- 


Fi-.  5. 


Fig.  6. 


ing  its  optical  center,  lay  off  four  points,  p.  p.  p.  p. ,  25  millimeters  from 
the  center,  as  shown  in  diagram  Fig.  5  : 

' '  Replace  the  lens  on  the  co7ivex  grinding  tool,  and  cut  the  lens  through 
the  indicated  dotted  right-angled  diameters  into  four  equal  parts.  This 
will  secure  four  quadrants,  with  ample  provisions  in  case  of  accident. 
Select  for  both  eyes  two  of  the  most  perfect  quadrants,  and  cut  from  them 
the  peripheral  segments  to  the  shape  indicated,  and  cement  the  wafers 
into  the  concave  7  D.  spheres,  with  their  thin  edges  down,  when  a  line 
LL  viewed  through  the  reading  lenses  will  appear  continuous  as  in  Fig.  6." 

It  is  obvious  that  the  diameter  of  the  lens  is  determined  by  adding  twice 
the  decentration  (25  X  2)  to  one  full  width  of  the  wafer,  (7  X  2)  which 
gives  64  millimeters. 

It  would  be  very  difficult  to  solve  this  simple  problem  so  easily  by  any 
other  means  than  the  prism-dioptry. 


WHY  STRONG  CONTRA-GENERIC  LENSES* 
OF  EQUAL  POWER  FAIL  TO  NEU- 
TRALIZE EACH  OTHER. 

RCYiscd  reprint  from  Annates  (V Oadistique  (English  Ed.)  November,  1895. 


Some  time  ago  Mr.  George  \V.  Wells,  President  of  the  American  Optical 
Company,  Soiithbridge,  Mass.,  requested  the  author  to  give  this  subject 
attention,  and,  as  it  contains  features  of  mutual  interest  to  oculists  and 
opticians,  it  has  been  considered  of  sufficient  importance  to  give  the  results 
of  this  investigation  publicity. 

In  practice  it  is  customary  to  determine  the  power  of  a  lens  by  what  is 
known  as  "neutralization."  It  is  here  proposed  to  show  why  it  is  that 
this  method  is  only  strictly  applicable  to  lenses  which  are  weaker  than  9 
dioptries.  The  power  of  a  lens,  as  is  well  known,  is  dependent  upon  three 
factors — the  radius  of  curvature,  index  of  refraction,  and  thickness  of 
glass.  The  latter  we  are  taught  to  consider  a  negligible  quantity,  since  it 
is  generally  infinitely  small  in  proportion  to  the  focal  distances  of  lenses 
which  are  used  in  spectacles.  This  is  only  justified  in  its  application  to 
coticave  lenses,  since  all  concave  lenses,  between  0.25  and  20  D.,  can  be 
made  of  the  same  infinite  thinness  in  the  center.  In  convex  lenses,  how- 
ever, we  meet  with  an  unavoidable  increase  in  thickness,  which  becomes 
of  sufficient  magnitude  in  lenses  above  8  D.  to  conflict  with  the  hypothesis 
referred  to.  When  the  element  of  thickness  is  considered,  we  have  the 
formula  for  bi-spherical  lenses  of  equal  curvatures  : 


F 


(^-D+^c^--)^^^-')' I. 


*  Lenses  of  opposite  character— convex  and  concave. 

1(51 


162 


WUY  STRONG  CONTRA-GENERIC  LENSES  FAIL  TO  NEUTRALIZE. 


wherein  r  is  the  radius  of  curvature,  n  the  index  of  refraction,  /^the  focus, 
and  e  the  thickness,  in  contra-distinction  to  the  formula  for  neglected  thick- 
ness, wherein 

r  =  2/^  («  —  1) II. 

It  is  therefore  evident  that  the  radius  of  curvature  will  be  a  different  one 
for  bi-convex  lenses,  in  which  thickness  is  considered,  from  that  of  bi-con- 
cave  lenses,  of  the  same  power,  having  no  appreciable  thickness. 


Fig.  1. 


The  accompanying  diagram,  Fig.  1,  representing  a  bi-convex  and  a  bi- 
concave lens  of  ideritical  curvatures,  clearly  shows  that  they  cannot 
optically  neutralize  each  other,  as  they  really  only  constitute  the  central 
portion  of  a  much  larger  periscopic  convex  lens,  and  which  our  imagina- 
tion can  construct  upon  the  dotted  lines  which  are  continued  to  their  in- 
tersections at  a  and  b. 

The  diagram  also  shows  that  the  power  of  the  imaginary  convex  menis- 
cus must  increase  with  an  increase  in  the  thickness  of  the  bi-convex  lens, 
because  the  anterior  and  posterior  surfaces  of  the  meniscus  will  be  ren- 
dered more  oblique  to  each  other  as  their  respective  centers  of  curvature, 
Cj  and  ^2,  are  separated  to  provide  for  an  increased  thickness.  The  nearest 
approach  to  neutralization  will  therefore  be  secured  when  the  centers  of 
curvature,  c^  and  c^,  are  as  close  together  as  possible,  thus  making  the  bi- 
convex lens  L  exceedingly  small  and  thin,  as  shown  in  Fig.  2. 

The  lenses  in  our  trial  cases  are,  however,  too  large  to  secure  even  this 
approximate  neutralization.     Their  diameter  of  necessity  determines  the 


WHY  STRONG  CONTRA-GENERIC  LENSES  FAIL  TO  NEUTRALIZE. 


163 


thickness,  vfhich  must  increase  with  the  power.  For  instance,  in  a  20  D. 
convex  trial-case  lens  of  3.5  centimeters  diameter  we  find  the  minimum 
thickness  to  be  0.75  centimeters.  If,  therefore,  in  Formula  I,  we  place 
e  =  0.75,  n  =  1.5,  and  F=  5  centimeters,  we  obtain  4.87  centimeters  as 
the  value  of  r,  whereas,  for  a  20  D.  concave  lens,  according  to  Forrriula 
II,  we  find  r  =  5  centimeters. 

As  the  radius  is  shorter  for  the  convex  than  for  the  concave  lens  of  the 
same  power,  it  is  evident  that  their  surfaces  will  actually  only  touch  in 
the  center,  as  exaggeratedly  shown  in  Fig.  3. 

Besides,  the  outer  surface,  s^  s^,  of  the  convex  lens  will  be  even  more 
oblique  relatively  to  the  outer  one,  s,  s,,  of  the  concave  lens,  so  that  these 
lenses  actually  form  the  center  of  a  stronger  convex  meniscus  than  shown 


\ 


Fig.  4. 


in  Fig.  1.  Or,  viewing  it  in  another  light :  Parallel  rays,  z,  which  are  in- 
cident to  the  concave  lens,  are  refracted  by  it,  passing  into  the  convex 
lens,  as  if  emanating  from  the  virtual  focal  point  /,  of  the  concave  lens, 
which  is  outside  of  the  focal  point,  /^,  of  the  convex  lens  in  Fig.  4. 

Such  rays,  e,  will  therefore  be  rendered  convergent,  instead  of  parallel, 
in  passing  out  of  the  convex  lens,  showing  that  neutralization  does  not 
exist. 

The  focal  distance,  in  infinitely  thin  lenses,  is  counted  from  a  single 
point,  s,  on  the  optical  axis  in  the  center  of  the  lens,  whereas  in  lenses  of 
appreciable  thickness  it  is  counted  from  the  posterior  principal  point,  h^^ 
within  the  lens.  In  a  bi-convex  lens  of  equal  curvatures,  made  of  glass, 
with  an  index  of  refraction  n  =  1.5,  it  has  been  demonstrated  that  the 
principal  points,  //,  and  h.,,  are  separated  by  a  distance  equal  to  one-third 


164  WHY  STRONG  CONTRA-GENERIC  LENSES  FAIL  TO  NEUTRALIZE. 

of  the  lens  thickness.*  It  is  therefore  obvious  that  the  focal  distance  of 
the  convex  lens  will  have  to  be  increased  by  at  least  one-third  of  the  lens 
thickness,  so  as  to  have  /^  and  /  coincide  for  the  purpose  of  effecting  neu- 
tralization. 

With  a  minimum  thickness  equal  to  0. 75  cm. ,  we  must,  consequently, 
add  0.25  to  the  focal  distance,  5,  making  5.25  the  focal  distance  of  the 
convex  lens.  This  corresponds  to  a  refraction  of  19.047  dioptrics.  Con- 
sequently, a  ip.o^y  convex  lens  of  o.y^  cm.  thickness  neutralizes  a  20  D. 
concave  lens  of  710  thickyiess.  To  be  more  accurate,  we  should  actually 
allow  for  one  millimeter  thickness  of  the  concave  lens.  This  would  result 
in  the  convex  lens  being  even  somewhat  weaker  than  19.047  dioptrics. 


Fig.  5. 


However,  according  to  Formula  I,  a  19.047  D.  convex  lens  of  0.75  cm. 
thickness  should  have  a  radius  r  =  5.121  cm.,  so  that  the  superposed 
neutralizing  lenses  would  actually  touch  each  other  at  their  edges,  instead 
of  at  the  center,  as  exaggeratedly  shown  in  Fig.  5. 

Furthermore,  any  additional  increase  in  the  thickness  of  the  convex 
lens  will  be  associated  with  an  increase  in  the  distance  Z/^/,  and  will 
therefore  call  for  a  corresponding  decrease  in  the  power  of  the  convex  lens 
to  produce  neutralization. 

Thus,  the  tendency  will  invariably  be  to  overestimate  the  power  of  the 
convex  lens,  when  an  effort  is  made  to  determine  its  power  by  that  of  the 
concave  lens  which  neutralizes  it. 

Calculation  shows  that  discrepancies  in  neutralization  varying  between 
0.25  and  1  D.  exist  in  the  entire  series  of  convex  lenses  between  9  and  20  D. 
It  consequently  also  follows  that  the  indiscriminate  addition  of  lenses,  as 
frequently  practiced  during  the  subjective  method  of  examination  of  ocular 
refraction,  is  not  permissible  for  lenses  of  high  power. 

•Miiller-Pouillet's  Lehrbucli  dcr  Physik,  page  160.    Braunschweig,  1894. 


WHY  STRONG  CONTRA-GENERIC  LENSES  FAIL  TO  NEUTRALIZE.  165 

In  other  words,  lenses  of  high  power  are  no  more  capable  of  being 
algebraically  combined  than  prisms  of  high  power,  a  fault  for  which  the 
practicability  of  the  prism-dioptry  was  so  severely  criticised.  The  same 
logic  therefore  applies  to  lenses  which  was  mentioned  in  the  author's  paper 
on  the  Prismometric  Scale,  to  wit : 

''The  desire  to  multiply  any  unit  in  optics  should  be  curbed  by  a 
knowledge  of  the  fact  that  all  the  fundamental  optical  laws  are  based  upon 
the  assumption  and  acceptance  of  values  of  limited  magnitude,  and  that 
there  is  therefore  apt  to  be  a  point  where  unreasonable  multiplication  of  an 
optical  unit  will  contradict  the  actually  existing  optical  phenomenon. ' ' 

The  general,  though  erroneous,  impression  that  the  entire  series  of  cor- 
responding contra-generic  lenses  should  neutralize  has  gained  such  credence 
that  lens  manufacturers  have  allowed  themselves  to  be  swayed  by  this 
popular  opinion,  and  as  a  result  have  adopted  the  principle  of  making  the 
convex  weaker  than  the  concave  lenses,  so  as  to  meet  the  demand  for 
neutralization.  As  has  been  shown,  a  20  D,  convex  lens  should  have  a 
shorter  radius  than  a  concave  one  of  the  same  power,  yet  examination  of 
any  trial  case  will  reveal  the  fact  that  the  reverse  is  the  case,  when  the 
surfaces  are  measured  by  a  suitable  gauge.  We  can,  however,  gain  no 
reliable   information   regarding  the   power   of   a  strong  convex  lens  by 


/ 

b     d 

Fig.  6. 

measurement  of  its  surfaces,  since  two  lenses  of  the  same  curvature,  but  of 
different  thickness,  will  not  be  of  the  same  power.  As  the  thickness 
increases,  the  power  will  diminish  for  one  and  the  same  curvature. 

This  is  shown  in  Fig.  6. 

The  incident  ray,  e,  is  refracted  by  the  anterior  surface  of  the  lens  ab  in 
the  direction  xf^^  and  by  the  posterior  surface  sX y  to  the  focus/.  If  the 
thickness  be  increased,  so  as  to  place  the  posterior  surface  at  cd,  then  xf^ 
will  be  refracted  by  the  posterior  surface  at  z  to  F,  parallel  to  j/,  since  the 
surface  cd  is  of  the  same  radius  as  ab. 


166   WHY    STRONG    CONTRA-GENEEIC    LENSES    FAIT.    TO    NEUTRALIZE. 

The  focal  distance  HJ^,  in  the  diagram,  is  not  appreciably  different 
from  h.J ;  in  fact,  the  difference  between  these  distances  scarcely  amounts 
to  0.05  cm.,  for  a  curvature  of  5  cm.  in  lenses  of  0.75  and  1  cm.  thicloiess. 
Nevertheless,  such  a  difference  would  be  appreciated  by  the  eye  in  neutral- 
izing. The  question  then  arises:  By  what  method  should  we  determine 
the  power  of  strong  lenses?  Indeed,  nothing  seems  to  remain  but  to  meas- 
ure them  by  a  focusing  screen  upon  a  graduated  bar,  care  being  taken  to 
count  the  focal  distance  from  the  posterior  principal  point  within  the  lens. 
Such  precaution  is,  however,  rarely  if  ever  taken.  While  concave  lenses 
can  be  similarly  measured,,  yet  this  is  somewhat  difficult,  and  not  entirely 
satisfactory.  The  lens  surface-measure  is  indeed  preferable  only  for  con- 
cave lenses,  since  they  are  of  negligible  thickness.  For  convex  lenses  above 
8  dioptrics  this  instrument  is  absolutely  unreliable. 

It  will,  however,  be  more  or  less  inconvenient  to  have  individually  dif- 
ferent methods  of  measurement  for  convex  and  concave  lenses.  Unless, 
therefore,  we  are  prepared  to  make  some  radical  change,  we  shall  do  well 
to  adhere  to  the  practice  of  neutralization,  bearing  in  mind  the  errors  wc 
commit  by  so  doing.  When  convex  lenses  stronger  than  8  dioptrics  are 
prescribed,  and  they  are  measured  by  neutralization  with  a  concave  lens, 
we  should  remember  that  they  are  always  weaker  than  the  dioptrics  indi- 
cated by  the  concave  lens.  In  short,  the  convex  lenses  in  our  trial  cases  are 
not  what  they  are  numhered  in  dioptries. 

So  long,  however,  as  manufacturers  are  agreed  that  the  convex  lenses 
they  produce  shall  neutralize  with  standard  concave  lenses,*  we  shall  at 
least  have  a  uniformity  which  will  insure  the  convex  lenses  in  spectacles 
being  duplicates  of  those  in  the  trial  case;  provided,  also,  that  the  lenses 
of  any  given  number  are  always  of  the  same  uniform  thickness.  This 
thickness,  in  every  instance,  should  be  the  least  which  can  be  given  to  the 
lens  of  3.5  cm.  standard  diameter. 


*Lenses  made  by  the  Ameiicaii  Optical  Compauy  are  ground  upon  this  principle  and  should 
be  tested  accordingly.  In  neutralizing  always  hold  the  convex  lens  next  to  the  eye.  If 
lenses  are  made  upon  the  opposite  principle,  they  will  not  neutralize  with  these. 


THE  ADVANTAGES  OF  THE  SPHERO-TORIC 

LENS. 

Revised  reprint  from  the  Ophthalmic  Record,  July,  1895. 


The  sphero-toric  lens  having  been  described  in  the  author's  treatise  on 
Ophthalmic  Lenses,  we  shall  here  proceed  to  show  the  various  advantages 
which  this  lens  possesses  over  its  sphero-cylindrical  equivalent  in  correct- 
ing astigmatism.  This  elucidation  will,  however,  be  confined  chiefly  to 
applications  of  the  toric  lens  in  cases  of  aphakia. 


In  Fig.  1,  Zrj  represents  a  plano-convex,  and  Z„  a  plano-concave  toric 
lens.  Fig.  2  represents  a  sphero-toric  lens  with  the  toric  surface  in  front, 
and  the  spherical  surface  behind. 

Similarly,  in  examining  the  refraction  in  aphakia,  it  is  customary,  as  in 
other  cases,  to  place  the  spherical  lens  in  the  groove  at  the  back  of  the 
trial-frame  and  therefore  nearer  to  the  eye,  with  the  cylinder  in  front. 
For  example,  in  a  case  involving 


-f  9  D.  sph. 


+  3.5  D.  cyl.  ax.  160^ 

167 


168  THE  ADVANTAGES  OF  THE  SPHERO-TORIC  LENS. 

the  result  is  obtained  by  placing  a  bi-convex  spherical  lens  of  +  9  D.  in 
the  trial-frame  behind  the  cylinder  +  3.5  D. 

The  optician,  hov/cver,  carries  out  this  prescription  literally,  by  making 
the  lens  \Yith  +  0  D.  spherical  on  one  side,  and  +  3.5  D.  cylinder  on  the 
other,  thereby  substituting  a  plano-convex  spherical  element  for  the  bi- 
convex one  used  in  the  trial  frame.  Furthermore,  in  mounting  the  lens, 
he  places  the  spherical  surface  farther  from  the  eye,  with  the  cylinder  in- 
side, so  that  the  spectacles  are  worn  with  the  surfaces  of  the  lenses  reversed 
in  respect  to  their  positions  before  the  eye  in  the  trial-frame. 

It  is,  therefore,  obvious  that  the  serious  error  is  committed  of  substi- 
tuting a  lens  which  involves  a  change  in  the  position  of  the  "nodal 
points ' '  of  the  entire  refracting  system,  thus  to  a  degree  impairing  th  ^ 
visual  acuity  obtained  by  the  spectacles,  as  compared  with  that  secured  by 
the  test  in  the  trial  frame.  Aside  from  this  we  have,  in  the  aforesaid 
sphero-cylindrical  lens,  the  unpleasant  j^henomenon  of  internal  reflection, 
caused  by  the  extreme  bulging  forward  of  the  convex  element,  whose  sur- 
face is  exposed  to  light  coming  from  other  directions  than  that  of  the  de- 
sired central  incident  beam. 

Every  ray  of  light,  R,  Fig.  3,  entering  the  convex  spherical  surface  in 
a  direction  coincident  with  its  radius  of  curvature,  will  enter  the  lens  un- 
refracted,  and  fall  upon  the  inner  surface  Cof  the  cylinder  on  the  other  side. 


Center  of  spherical   y-«*'''   |/\      1     *              *~^''-"-*-^-'it-.fv»  Center  of  cylindrical 

curvature.  «j^;---.K..-x^.-  -.— ^ — ..•.;?.*vs^  curvature. 


Fig.  3. 

Under  favorable  conditions  it  will  be  reflected  from  this  back  to  the 
inner  anterior  surface  of  the  lens,  there  again  undergoing  reflection,  and 
so  on,  until  it  becomes  dissipated  within  the  lens  as  diffuse  light.  The 
dotted  lines  in  the  diagram  represent  the  radii  of  curvature,  dividing  the 
angles  of  incidence  and  reflection,  which  are  equal  at  the  points  of  impact 
on  the  surfaces  within  the  lens. 


THB  ADVANTAGES  OF  TUE  SPHERO-TORIC  LENS.  169 

The  patient  wearing  such  spectacles  is  therefore  compelled  to  look,  as  it 
were,  through  a  self-luminous  medium,  which  tends  to  interfere  with  the 
direct  central  incident  beam  of  light  passing  through  to  his  pupil.  Again, 
in  this  form  of  lens  the  aberration  is  greater  for  all  excursions  of  the  eye 
where  the  visual  axis  passes  through  the  lens  at  points  not  coinciding  ex- 
actly with  the  optical  axis  of  the  lens. 

These  facts  were  clearly  exemplified  in  a  case  which  came  to  the  au- 
thor's notice  in  1889.  The  lens  had  been  made  of  the  sphero-cylindrical 
form  and  povrer  as  above  stated.  The  patient  complained  of  inability  to 
correctly  judge  distances  in  looking  down,  and  of  an  annoying  glare  of  light 
which  he  felt  was  within  the  lens  itself  when  worn  out  of  doors,  or  in  a 
strong  light.  He  thought,  if  the  glare  could  be  removed  he  would  be  en- 
abled to  see  better.  In  fact  he  had  made  the  experiment  of  shading  the 
lens  by  sighting  through  the  partially  closed  hand  held  closely  before  it, 
and  had  noticed  less  inconvenience  from  the  glare. 

The  sphero-cylindrical  correction,  then  worn  by  him,  gave  him  scarcely 
better  than  6/9  of  vision,  whereas  the  test  lenses  in  the  trial  frame,  by  the 
author's  test,  gave  him  the  somewhat  unusual  acuteness  of  6/6. 

It  occurred  to  the  author  that  the  interior  reflections  could  be  avoided 
by  constructing  the  lens  with  less  curvature  on  the  anterior  surface.  There- 
fore, a  lens  having  a  toric  surface  on  the  anterior  side,  and  a  spherical  sur- 
face on  the  other  was  suggested  as  follows  :  +  6  D.  sph.  combined  with  a 
toric  surface  of  +6.5  D.  ax.  160°  by  +  3  D.  ax.  70°,  in  place  of  +  9  D. 
sph.  O  +  3.5  D.  cyl.  ax.  160°. 

Before  proceeding  to  consider  the  equivalence  of  these  lenses,  it  is  sug- 
gested that  all  thought  of  a  simple  cylindrical  form  should  be  dispelled 
from  the  mind  when  the  toric  surface  is  referred  to.  The  subject  will  be 
more  easily  understood  by  dealing  only  with  the  principal  refracting 
meridians,  without  regard  to  the  other  meridians  of  curvature,  which  give 
form  to  the  toric  surface. 

Referring  to  the  lens,  +  9  D.  sph.  Q  +  3.5  D.  cyl.,  it  is  evident  that  the 
meridian  of  least  refraction  is  9  D.,  and  the  meridian  of  greatest  refraction 
is  12.5  D.,  see  m  and  il/in  the  perpendicular  planes  of  Fig.  4. 

The  extreme  spherical  curvature,  9  D.,  on  the  anterior  surface  may  be 
lessened  to  any  desired  degree,  provided  the  loss  of  refraction  is  compen- 


170 


THE  ADVANTAGES  OF  THE  SPHERO-TORIC  LENS. 


sated  for  by  adding  it  to  the  opposite  cylindrical  side  of  the  lens,  in  which 
case  the  simple  cylindrical  surface  would  necessarily  be  replaced  by  a  sur- 
face of  astigmatic  refraction,  a  toric  surface,  having  the  required  focal  in- 
terval of  3.5  D.  To  obtain  such  an  equivalent  lens  of  thinnest  and  best 
form,  it  is  only  necessary  to  divide  the  original  meridian  of  greatest  re- 
fraction, M=  12.5  D.,  in  two  parts,  as  nearly  equal  as  possible;  say,  6  D, 


M-12.5  Mr  12.5 


Fig.  4. 


for  the  back  surface,  and  6.5  D.  for  the  front  surface,  which,  together, 
give  12.5  D.  for  the  newly-created  bi-convex  meridian  of  greatest  refraction, 
M ,  Fig.  4,  in  the  desired  equivalent  lens. 

Taking  6  D.  as  the  posterior  spherical  surface,  it  is  necessary  to  com- 
bine it  with  3  D.  on  the  anterior  surface  to  secure  9  D.  as  the  meridian  of 
least  refraction,  m^,  in  the  new  lens.  Consequently  6.5  D.  and  3  D.,  re- 
spectively, represent  the  refraction  of  each  of  the  principal  meridians  of 
the  anterior  toric  surface,  which,  when  combined  with  the  6  D.  posterior 
spherical  surface,  will  fulfil  all  the  requirements  of  equivalence  as  shown 
in  the  diagrams. 

The  sphero-toric  lens  referred  to  was  set  in  a  spectacle-frame  with  the 
toric  surface  outward,  and  its  meridian  of  least  refraction  at  160°,  so  that 
the  refracting  elements  occupied  the  same  positions  as  the  lenses  in  the 
trial  frame. 

The  spectacles  have  been  worn  by  the  patient  for  the  past  five  years, 
with  complete  relief  from  all  the  disagreeable  phenomena  mentioned.  It 
may  also  be  stated  that  his  visual  acuteness  with  the  new  lens  is  slightly 
better  than  6/6. 

For  reading,  the  patient  required  -f  12  D.  sph.  Q  +  3.5D.  cyl.  160°  in 
the  trial-frame,  which  was  given  him  in  the  form  of  a  sphero-toric  lens  : 
-f  8D.  sph.  C  toric  surface  of  -f  7.5  D.  ax.  160°  by  +  4  D.  ax.  70°. 


THE  ADVANTAGES  OF  THE  SPHERO-TORIC  LENS.  l7l 

In  addition  to  the  advantages  mentioned,  the  patient  is  saved  the  annoy- 
ance of  wearing  uncomfortahly  heavy  lenses,  which  also  attract  attention. 
In  every  instance  where  the  autlior  lias  applied  the  sphero-toric  lens  it  has 
given  entire  satisfaction,  though  in  none  of  the  cases  has  the  visual  acute- 
ness  been  so  perfect  as  in  the  one  just  cited. 

The  use  of  the  sphero-toric  lens  is  by  no  means  confined  to  cases  of 
aphakia,  since  equally  good  results  can  be  secured  by  its  use  in  high 
degrees  of  compound  myopic  astigmatism,  especially  where  the  cylindrical 
corrections  are  weak  in  comparison  to  the  high  spherical  curvatures 
involved. 

Furthermore,  tlie  sphero-toric  lens  is  also  capable  of  being  given  a 
periscopic  form,  and  which,  if  desired,  may  be  made  quite  as  globular  as 
the  so-called  coquille  glass.  In  this  form,  especially,  it  affords  the  advan- 
tage, when  placed  before  the  eye,  of  allowing  its  peripheral  area  to  be 
brought  nearer  to  and  more  concentric  with  the  eye-ball  than  is  possible 
with  the  sphero-cylindrical  equivalent;  so  that,  for  all  ordinary  motions 
of  the  eye,  the  visual  axis  will  be  less  oblique  to  the  inner  surface  of  the 
lens.  This,  and  the  consequent  absence  of  reflection  from  the  inner  con- 
cave surface,  also  give  this  form  of  sphero-toric  lens  a  wider  and  more 
natural  field  of  vision  than  is  obtained  by  the  ordinary  sphero-cylindrical 
equivalent.  This  feature  is  appreciated  to  a  marked  degree  by  those  who 
wear  them  in  shooting,  or  at  billiards,  tennis,  golf,  etc. 

A  further  commendable  feature  is  that  sphero-toric  lenses  of  high  power 
can  be  made  very  much  thinner,  and  consequently  of  less  weight  than  is 
ordinarily  possible  in  such  cases.  They  are  also  frequently  advantageous 
where  cemented  bi-focal  lenses  are  required,  since  the  segments  for  reading 
are  inclined  at  an  angle  more  closely  approaching  position  for  perpen- 
dicular incidence  of  the  visual  axes  when  looking  downward.. 

Although  sphero-toric  lenses  are  considerably  more  expensive  than  others, 
it  is  fair  to  predict  a  decided  increase  in  their  application  by  com- 
petent optical  practitioners,  as  soon  as  the  aforesaid  advantages  shall  have 
become  more  widely  known. 


THE  IRIS,  AS  DIAPHRAGM  AND 
PHOTOSTAT* 

Revised   reprint   from    the    "Annals   of   Ophthalmology   and   Otology,"    October,    1895. 


Under  this  title  it  is  proposed  to  inquire  into  the  value  of  sub-decimals 
of  the  lens-dioptry  in  ametropia.  The  subject  has  been  frequently  dis- 
cussed, and  again,  at  considerable  length,  at  a  meeting  of  the  American 
Medical  Association,  in  San  Francisco,  Cal.,  1894,  with  the  result  that 
''low  degree  lenses"  are  now  generally  conceded  to  have  a  noteworthy 
therapeutic  effect;  though  no  scientific  reason  has  been  given,  and  simply 
because  the  physical  laws  involved  have  never  even  been  mentioned.  While 
unanimity  of  opinion  of  this  sort  may  be  exceedingly  satisfactory  from  a 
medical  point  of  viev/,  yet  it  only  circumstantially  corroborates  that 
relevant  scientific  argument  which  should  properly  also  embrace  the  fol- 
lowing important  considerations. 

In  every  compound  lens-system  we  are  met  with  the  necessity  of  provid- 
ing against  spherical  aberration.  This  is  accomplished,  in  the  construction 
of  optical  instruments,  by  introducing  an  annular  disk,  of  calculated 
diameter,  known  as  the  diaphragm,  which  is  suitably  placed  between  the 
lenses  to  exclude  peripheral  rays.  If  the  proper  diaphragm  be  replaced  by 
one  of  smaller  aperture,  we  increase  the  definition,  but  diminish  the  exteiit 
of  field  and  illumination.  A  larger  aperture  will  increase  illumination  and 
field,  but  definition  will  be  impaired,  on  account  of  the  aberration  thus 
allowed. 

The  aperture  of  the  diaphragm  must  therefore  have  a  definite  and 
specific  diameter  for  every  optical  instrument,  if  we  arc  to  secure  maximum 
definition  and  illumination,  without  aberration.    The  proper  diaphragm  is 

'Photostat,  Greek,  (^is  ((Jwt-),  light,  +  o-Tards,  verbal  adjective  of  iardrai,  stand  —  an  automatic 
light  regulator  (suggested  by  the  author). 

173 


174 


THE    IRIS,    AS    DIAPHEAGM    AND    PHOTOSTAT. 


therefore  one  of  the  most  important  and  indispensable  parts  of  every  com- 
])0uud  dioptric  system.  The  human  eye  is  such  a  system,  and  is  provided 
with  its  diaphragm — the  iris.  In  the  eye,  which  is  a  dynamic  apparatus 
given  to  variations  of  power,  a  fixed  diameter  of  pupil  would  fail  to 
theoretically  fulfill  the  requirements.  When  the  eye  is  in  a  state  of  accom- 
modation, it  becomes  a  stronger  refracting  system,  and  therefore  needs  a 
smaller  aperture  of  diaphragm,  hence  the  pupil  contracts.*  Yet,  Helm- 
holtzf  says:  "A,  von  Graefe  observed  in  an  eye  from  which  he  had 
removed  the  iris  by  operation  that  the  normal  range  of  accommodation 
was  still  present,  and  also  that  the  changes  in  the  anterior  curvature  of  the 
lens  could  still  be  observed."  He  concludes :  "The  iris  does,  therefore, 
not  play  an  important  role  in  accommodation."  (Lit.  trans.)  LandoltJ 
expresses  the  same  opinion.  So  far  as  the  above  noted  measurements  are 
concerned,  such  conclusion  may  be  quite  correct,  yet  if  construed  in  its 
broadest  sense  it  discountenances  the  value  of  the  iris  as  a  diaphragm 
entirely. 

It  is,  nevertheless,  universally  admitted  that  the  iris  does  act  inde- 
pendently of,  and  simultaneously  with  accommodation.  §  "When  acting 
independently  of  accommodation,  the  iris  is  known  to  behave  as  a  highly 
sensitive  photostat,  through  regulating  the  volume  of  light  upon  the  retina 
to  such  a  degree  as  shall  be  most  agreeable  to  our  light-perceptive  sense. 

A  most  subtile  and  sjoichronous  balance,  between  retinal  perception, 
uveal  stimulus,  and  iritic  response,  must  therefore  exist,  if  the  iris  is  to 
perform  its  functions  simultaneously  as  diaphragm  and  photostat. 

An  endeavor  will  here  be  made  to  support  the  hypothesis  that  a  dis- 
turbed equilibrium  of  these  functions  is  probably  the  cause  of  asthenopia  in 
low  degrees  of  ametropia.  From  a  strictly  optical  point  of  view  every  eye 
of  the  same  refraction,  other  things  being  equal,  should  have  a  pupil  of  the 
same  diameter — one  suited,  by  calculation,  to  exclude  peripheral  aberra- 
tion, while  securing  the  greatest  tolerable  illumination.    This,  however,  is 


*  In  fact,  it  was  at  one  time  supposed  that  contraction  of  the  pupil  was  the  only  means  by 
which  the  eye  adapted  itself  for  near  vision.  Helmholtz,  "Physiologische  Optik,"  page  151, 
Hamburg  and  Leipzig,  1886. 

t Helmholtz,   "Physiologische  Optik,"  page  138. 

t  Ijandolt,    "Refraction  and  Accommodation  of  the  Eye,"  page  lo4,  Philadelphia,   1886. 

§  "Movements  of  the  iris  are  nevertheless  associated  with  accommodation;  they  are  governed 
by  the  same  nerves  as  the  latter,  so  that,  until  the  mechanism  of  accommodation  is  better 
understood,  a  direct  relation  between  them  may  not  be  looked  upon  as  being  improbable." 
(Lit.  trans.)    Donders,    "Refraction  and  Accommodation,"  page  485,  Wein,   1866. 


THE    IKIS,    AS    l)IAPIIRA(iM    AND    PHOTOSTAT.  175 

not  known  to  be  the  case,  nor  has  the  author  found  that  any  one  has  ever 
calculated  what  tlio  diameter  of  the  pupil  should  be  for  any  given  schematic 
eye.  Listing  has  calculated  a  table  showing  the  changes  in  diameter  of 
the  diffusion  circles  upon  the  retina  which  arise  through  efforts  of  accom- 
modation in  a  schematic  eye  having  a  pupil  of  4  mm* 

We  have  thus  far  been  content  to  Imow  that  pupils  differ  in  size  in  dif- 
ferent persons.  There  must,  however,  be  a  limit  to  the  maximum  diam- 
eter of  the  pupil,  if  aberration  is  to  be  excluded,  and  if,  for  any  reason, 
the  pupil  is  prevented  from  contracting  to  at  least  this  limit,  we  shall  have 
aberration,  even  in  the  emmetropic  eye. 

This  is  exaggeratedly  shown  in  Fig.  1,  in  which  the  central  incident 
rays,  cc,  focus  at  f  upon  the  retina,  while  the  peripheral  rays,  pp,  produce 


Pig.  1. 


thereon  an  area  of  diffusion,  f  ab,  and  which,  to  all  practical  purposes,  would 
be  equally  as  effective  in  impairing  vision  as  a  low  degree  of  myopia,  hav- 
ing its  intra-ocular  focus  anywhere  between  the  retina  and  f^.  In  fact,  it 
is  questionable  whether  the  eye  can  discriminate  between  images  which  are 
impaired  by  peripheral  aberration  and  those  which  are  illy  defined  through 
slight  errors  of  refraction.  The  following  experiment  will  serve  to  illus- 
trate this:  By  placing  a  1  D.  convex  lens  before  the  emmetropic  eye,  it 
is  practically  rendered  myopic  for  distance,  the  letters  of  the  test-card  at 
6  m.  becoming  indistinct,  with  a  probable  reduction  in  the  visual  acute- 
ness  to,  say,  |.  If  the  lens  be  now  covered  with  a  pin-hole  disk,  normal 
acuteness  of  vision  will  be  re-established,  with  no  other  appreciable  differ- 
ence than  that  the  field  and  illumination  are  less.  We  may  therefore  con- 
sider the  peripheral  rays,  here  accompanying  the  increased  refraction,  as 
aberrative  rays  in  respect  to  the  enclosed  central  incident  beam,  so  that  an 


*  Helmholtz,  "Physiologische  Optik,"  page  127. 

t  For  purposes  of  lucid  illustration,  the  diffusion  areas  in  all  of  the  diagrams  are  greatly 
exaggerated. 


176  THE    IRIS,    AS    DIAPHRAGAI    AND    PHOTOSTAT. 

eye  capable  of  contracting  its  pupil  to  the  same  extent  would,  in  part, 
similarly  correct  its  error  of  refraction. 

This  is  undoubtedly  one  reason  why  errors  of  refraction  of  the  same 
degree  are  not  accompanied  by  the  same  diminution  of  visual  acutenoss. 
The  myope  of  1  D.,  with  small  pupils,  without  glasses,  will  probably  have 
better  vision  than  the  myope  of  1  D.  with  much  larger  pupils.  Within 
certain  limits,  peripheral  aberration  and  anomalies  of  refraction  are  anal- 
ogous in  destroying  definition  of  the  image.  A  slight  error  of  refraction, 
with  large  pupils,  may  produce  diffusion  images  equally  as  pronounced  as 
a  considerable  refractive  error  with  small  pupils. 

Asthenopia  is  therefore  quite  as  apt  to  be  experienced  on  ac- 
count of  the  size  of  the  pupil  as  it  is  on  account  of  the  error  of 
refraction. 

This  should  explain  why  it  is  that  many  persons,  having  small  pupils, 
endure  a  considerable  error  of  refraction  without  inconvenience,  whereas 
others,  with  large  pupils  and  small  errors  of  refraction,  are  afflicted  with 
asthenopia. 

Again  reverting  to  Fig.  1,  the  larger  the  pupil  the  greater  will  be  the 
zone  of  peripheral  aberration  and  its  correlated  diffusion-area,  ah.  In  fact 
"the  peripheral  aberration  upon  the  optical  axis  is  known  to  increase,  not 
only  in  proportion  to  the  square  of  the  aperture,  but,  also  pari  passu  with 
the  refraction"  (physical  law),  so  that  we  should  have  greater  diffusion 
circles  upon  the  retina,  when  the  ciliary  muscle  is  brought  into  action, 
even  in  emmetropia,  to  correct  the  peripheral  aberration  which  impairs  the 
sharp  definition  at  /.  The  only  stimulus  which  could  assist  in  correcting 
the  aberration  in  this  case  would  be  that  M'hich,  imparted  to  the  iris  from 
the  retina,  would  cause  the  pupil  to  contract  sufficiently  to  exclude  periph- 
eral rays.  In  here  speaking  of  the  retina,  we  of  course  take  for  granted  its 
highest  state  of  ph3'siological  development.  The  question  then  arises:  Is 
such  retinal  stimulus  imparted  to  the  iris  in  low  degrees  of  ametropia,  in- 
dependent of  accommodation,  without  increased  light  intensity?  If  there  is 
such  independent  action  on  the  part  of  the  iris,  ineffectual  efforts  of  the 
ciliary  muscle  to  correct  impaired  vision  may  be  followed  by  a  contraction 
of  the  pupil  necessary  to  shut  out  the  peripheral  rays.    As  to  this,  let  us 


THE    IRIS,    AS    DIAPHRAGM    ANT)    PHOTOSTAT. 


177 


invoetigatc  the  relation  which  should  exist  between  the  iris  and  accommoda- 
tion in  the  hyperopic  eye. 


Fig.  2. 


In  this.  Fig.  2,  the  central  rays,  cc,  are  focused  behind  the  retina  at  f, 
the  peripheral  rays  crossing  at  /^  and  producing  the  diffusion-area  ah.  In 
facultative  hyperopia  there  will  be  accommodation  sufficient  to  bring  /  for- 
ward to  the  retina.  With  this  increased  refraction,  however,  the  pupil  re- 
maining the  same,  /^  will  recede  from  the  retina,  Avith  a  corresponding 
increase  in  the  size  of  the  diffusion-area  ah*    It  is  therefore  evident  that. 


Fig.  3. 


if  increased  aberration  is  to  be  avoided,  a  normal  pupil  must  contract  con- 
currently with  the  accommodation.  This,  generally  speaking,  is  knoAvn  to 
be  the  case.  If,  as  in  Fig.  3,  the  hyperopia  is  of  low  degree,  with  exces- 
sively large  pupil,  we  shall  have  a  comparatively  small  central  area  of  dif- 
fusion, due  to  the  refractive  error,  covered  by  a  much  larger  area  of  diffu- 
sion and  illumination,  ah.  The  slightest  effort  of  accommodation  would 
tend  to  sustain  or  increase  this  discrepancy.  It  therefore  follows,  if  the 
aberration  is  to  be  abolished,  that  the  iris  must  receive  an  increased  stim- 
ulus to  bring  about  a  contraction  of  the  pupil,  in  excess  of  that  which  is  con- 
currently associated  with  accommodation,  and  that,  too,  for  every  degree  of 


♦Listing's  table  shows  that  the  diffusion  circles  upon  the  retina  increase   more  rapidly  as 
the  object  approaches  the  eye  at  short  range.     Helmholtz,    "Physiologische  Optik,"  page  128. 


178  THE    IRIS,    AS    DIAPHEAGM    AND    PHOTOSTAT. 

light  intcnsUy.  Were  this  not  the  case,  vision  at  a  distance,*  with  exces- 
sively large  pupils,  would  be  impaired  by  aberration  under  all  circumstances. 

The  additional  stimulus  to  contraction  is  undoubtedly  due  to  the  increased 
area  of  illumination  above  mentioned.  This  would  seem  to  imply  that  the 
contraction  of  the  pupil  not  only  resj^onds  to  the  light  intensity'  (quality) 
but  also  to  its  area  (quantity)  upon  the  retina. 

It  is  also  evident  that  the  impairment  of  vision  should  be  ascribed  to  that 
factor  causing  the  largest  area  of  diffusion  upon  the  retina.  The  larger  the 
pupil,  the  more  will  the  peripheral  aberration  predominate  over  that  which 
is  produced  in  the  center  by  a  low  degree  of  refractive  error. 


Fig.  4. 


By  placing  the  lens  before  the  eye  which  corrects  the  hyperopia  we  in- 
crease the  refraction,  thus  eliminating  the  diffuse  central  image,  but  at  the 
same  time  increasing  the  peripheral  aberration,  and  therefore  also  the  area 
of  illumination.  Fig.  4.  If  the  pupil  contracted  only  in  'proportion  to  the 
consequent  increased  light  stimulus  there  would  still  remain  the  original 
diameter  of  diffusion  area.  As,  however,  correction  of  the  refractive  error  by 
the  lens  improves  vision,  and  relieves  asthenopia,  being  tacit  proof  that  the 
aberration  is  dispelled,  it  is  evident  that  the  pupil  must  contract  more  thsin 
in  proportion  to  the  aforesaid  light  stimulus.  Is  it  not  then  probable  that 
the  pupil  contracts  more  freely  when  accommodation  is  relaxed  ? 

In  controverting  this,  it  would  be  necessary  to  refute  the  following  fact 
pertaining  to  combined  kinetic  energies : 

When  accommodation  is  in  force,  the  iris  is  known  to  be  carried  for- 


*  In  accommodation,  with  a  standard  light  placed  behind  the  plane  of  the  eyes,  and  an  ap- 
proach to  them  of  the  paper  upon  which  the  test-type  is  printed,  the  illumination  upon  the 
paper  increases  in  the  inverse  proportion  to  the  square  of  the  reduced  distance  between  the 
light  source  and  the  test-object.  The  illumination  also  varies  directly  as  the  cosine  of  the 
angle  of  incidence  upon  the  illuminated  surface.     (Physical  law  of  Photometry.) 


THE    IRIS,    AS    DIArilRAGM    AKD    I'JIOTOSTAT.  179 

ward,*  by  pressure  from  the  anterior  surface  of  the  lens,  wliich  lias  become 
more  strongly  curved.  Such  lens-pressure,  the  im  remaining  inactive, 
would  tend  to  increase  the  diameter  of  the  pupil.  On  this  account,  greater 
efforts  of  the  sphincter  will  be  necessary  to  counteract  this  action  of  the 
lens-surface,  when  accommodation  is  present,  Fig.  5,  than  it  would  with 
relaxed  accommodation,  Fig.  G. 

For  normal  conditions  of  innervation  the  sphincter  is  known  to  more 
than  overcome  such  action  on  the  part  of  the  lens  in  accommodation.  Fig.  5. 


Fig.  5.  Fig.  6. 


If,  therefore,  our  hypothesis  is  correct,  we  have  found  a  reason  why  low 
degree  lenses  are  of  so  much  benefit  in  slight  hyperopia,  and  congeneric 
astigmatism.  Furthermore,  we  are  justified  in  assuming  that  the  sphincter 
in  large  pupils  does  not  always  adequately  respond,  while  accommodation  is 
in  force,  especially  in  cases  where  the  optical  error  is  so  slight  as  a  quarter 
dioptry,  from  the  fact  that,  in  the  majority  of  such  cases,  the  patients  are 
young,  and  often  possess  amplitudes  of  accommodation  varying  between 
6  and  14  dioptrics. 

Patients  with  such  accommodation  have  so  much  of  it  in  reserve,  even 
when  using  the  eyes  in  proximity,  that  their  asthenopia  can  scarcely  be 
ascribed  to  an  overtaxed  ciliary  muscle.  Are  we  not  then  justified  in  at- 
tributing it  to  possible  fatigue  of  the  iris,  resulting  from  its  involuntarily 
prompted,  though  futile,  efforts  to  exclude  peripheral  aberration,  because 
of  the  sphincter's  inability,  for  some  reason,  to  contract  sufficiently? 

It  is  not  recorded  that  a  disproportion  of  the  pupils  to  the  dioptric  system 
of  the  eyes  does  ever  exist  physiologically,  but  there  are  many  conditions 


Helmholtz,    "Physiologische  Optik,"   page  131,  Wien,   188G. 


180  THE    lEIS,    AS    DIAPHRAGM    AND    PHOTOSTAT. 

of  the  nervous  system  which  produce  immoderate  dilatation  of  the  pupils. 
Such  dilatation,  while  it  lasted,  would  tend  to  oppose  the  normal  association 
between  refraction  and  the  correlated  size  of  the  pupil. 

In  those  cases  of  normal  pupil,  where  the  perceptive  qualities  of  the 
retina  are  good,  and  the  error  of  refraction  is  slight,  retinal  stimulus  will 
prompt  contraction  of  the  pupil  sufficient  to  exclude  aberration.  Is  it  not 
probable  that,  in  some  cases  with  large  pupils,  protracted  efforts  of  this 
kind  Avould  result  in  fatigue  of  the  iris?  Might  not  prolonged  ineffectual 
efforts  of  the  iris  to  regain  equilibrium  between  its  functions,  as  diaphragm 
and  photostat,  account  for  asthenopia?  Or,  to  put  it  in  another  way: 
Could  not  that  prolonged  effort  of  the  sphincter,  which  would  have  to  be 
in  excess  of  the  normal  qualitative  and  quantitative  light  stimulus,  to  correct 
aberration,  produce  asthenopia? 

It  need  not  follow  that  the  iris  is  incapable  of  temporarily  contracting 
even  to  a  greater  extent  than  is  necessary  for  the  above  purpose.  This  is 
demonstrated  by  the  extreme  contraction  of  which  the  pupil  is  generally 
capable  when  exposed  to  intense  light,  and  the  eye  is  in  its  static  state  of 
refraction. 

In  hyperopes,  we  generally  ascribe  the  cause  of  asthenopia  to  fatigue  of 
the  ciliary  muscle,  owing  to  its  efforts  to  exclude  the  error  of  refraction 
by  accommodation.  The  same  cannot  be  said  of  myopes,  whose  use  of 
accommodation  for  such  purposes  would  only  render  them  deplorably  more 
myopic.  Their  asthenopia  can  certainly  not  be  ascribed  to  ciliary  fatigue. 
Some  myopes,  however,  endeavor  to  improve  their  vision  by  compressing 
the  eyelids,  which  means  that  they  thereby  modify  the  pupils  to  exclude 
peripheral  rays,  and  the  aberration  which  is  heightened  by  the  myopia. 
In  low  degrees  of  myopia  and  congeneric  astigmatism,  however,  modifica- 
tion of  the  pupils,  by  compression  of  the  eyelids,  is  not  sufficiently  delicate 
to  exclude  aberration,  without  too  great  a  sacrifice  of  iUumination.  Such 
patients  are  therefore  more  apt  to  apply  for  relief  from  glasses,  than  those 
who  help  themselves  by  compression  of  the  eyelids,  provided  this  is  unac- 
companied by  asthenopia.  In  the  former  cases,  we  are  to  suspect  that  the 
relief  sought  is  freedom  from  peripheral  aberration.  The  latter  also 
aggravates  photophobia,  which  is  a  symptom  frequently  complained  of  in 
Buch  cases. 

The  improvement  in  vision,  which  the  myope,  of  low  degree,  with  large 
pupils,  secures  by  the  lenticular  correction,  is  practically  due  to  the  fact  that 


THE    IKIS,    AS    DIAPHRAGM    AND    PHOTOSTAT.  181 

the  peripheral  aberration  is  decreased,  through  reduced  refraction  obtained 
by  the  concave  lens  in  front,  Fig.  7. 

The  rays  emitted  from  the  concave  lens  enter  the  pupil  with  a  divergence 
•ounteracting  the  excessive  convergence  of  the   rays   which   are   imper- 


Fig.  7. 


fectly  focused  by  the  crystalline  on  the  retina  behind  f.  The  peripheral 
diffusion  area,  ah,  may  not,  however,  always  be  in  such  proportion  to  the 
central  diffusion  area  as  to  be  fully  corrected  by  the  lens  which  corrects 
the  refractive  error  in  the  center.  Should  it,  in  the  case  of  a  larger  pupil, 
be  greater,  the  patient  would  merely  then  select  a  stronger  lens,  having  its 
proper  effect  upon  peripheral  rays,  while  its  tendency  would  also  be  to  over- 
correct  the  myopia.  For  a  low  degree  of  myopia  this  would  scarcely  be 
appreciable,  since  very  little  difference  in  refraction  is  experienced  in  the 
actual  centers  between  lenses  of  a  quarter  and  a  half  dioptry. 

In  those  cases  where  the  qvxirter-dioptry  lens  seems  to  relieve  asthenopia 
it  will  generally  he  found  that  the  pupUs  are  comparatively  large.  This  is 
especially  noteworthy  in  those  cases  where  simple  myopes  of  low  degree 
are  benefited  by  wearing  their  weak  distance  corrections  for  reading,  and 
which  can  serve  no  other  needful  purpose  than  to  eliminate  peripheral 
aberration. 

So  far,  we  have  no  means  of  ascertaining  the  size,  or  that  variation  of 
the  pupil  which  is  necessary  to  establish  the  proper  harmony  between 
refraction,  accommodation,  illumination,  and  freedom  from  aberration. 
The  intuitive  discrimination,  which  accompanies  experience,  is  at  present 
our  only  guide. 

In  refractive  errors  of  low  degree,  which  are  relieved  by  lenticular  cor- 
rection, the  retinal  perception  is  usually  also  very  keen,  thus  increasing 
stimulus  to  contraction  of  the  sphincter,  while  the  correction  in  such  cases 
frequently  improves  vision  to  f  ,  which  is  far  above  normal. 


182  THE    IKIS,    AS    DIAPHEAGM    AND    I'UOTOSTAT. 

!^  The  larger  the  pupil,  the  more  pronounced  will  be  the  improvement  in 

visual  acuteness  obtained  by  low-degree  corrections.  The  quarter-dioptry 
lens  rarely  proves  of  l^enefit  when  the  pupils  are  small. 

Again,  patients  frequently  wear  such  glasses  for  a  time,  relieving  their 
asthenopia,  and  ultimately  lay  them  aside,  without  feeling  the  necessity  of 
their  further  use.  Examination  will  nevertheless  reveal  the  fact  that 
the  optical  error  has  not  changed.  Why  then  should  asthenopia  exist  at 
one  time,  and  not  at  another,  for  an  invariable  hypermetropic  astigmatism 
for  instance,  if  the  fatigue  in  the  first  instance  had  only  been  due  to  that  of 
the  ciliary  muscle  ? 

Closer  examination,  however,  will  frequently  sliow  that  the  pupils  appear 
to  be  smaller  at  the  time  the  patient  has  discarded  his  glasses  than  when 
they  were  prescribed.  The  pupil  being  the  only  member  seeming  to  have 
undergone  a  change,  are  we  not  justified  in  suspecting  the  iris,  by  reason 
of  disturbed  innervation,  as  having  been  at  least  implicated  in  the  cause  of 
asthenopia  ? 


fy.\<i 


THE  TYPOSCOPE. 

Revised  reprint  from  The  Keystone,   1897. 


It  is  commonly  understood  that  visual  acuteness  depends  upon  the  per- 
ceptive functions  of  the  retina,  as  well  as  upon  the  size  of  the  image  pro- 
jected upon  it,  and  which  is  limited  by  the  visual  angle  subtended  by  the 
object  at  the  nodal  point  within  the  eye.  To  the  exclusion  of  all  other 
considerations,  the  ability  to  discern  objects  therefore  primarily  depends, 
first,  upon  contrast  in  light  intensity,  involving  also  the  color  sense;  and, 
second,  upon  the  size  of  the  object  viewed.  Therefore,  the  greater  the 
contrast  between  the  color  of  the  object  and  the  background,  the  more  readily 
will  an  object  of  any  given  size  be  distinguished.  Thus  it  is  frequently 
observed  that  a  visual  acuteness  of  f  ,  with  diminished  illmnination,  is 
raised  to  f  with  a  maximum  illumination,  as  a  result  of  heightened  con- 
trast between  the  type  and  its  background.  In  cases  of  ametropia  and  ) 
amblyopia  it  is,  however,  also  frequent  that  increased  illimiination  reduces 
the  definition,  owing  to  a  superabundance  of  extraneous  light,  which  serves 
\^o  reduce  the  contrast  within  the  polar  field  of  fixation.  In  optical  instru- 
ments it  is  found  practicable  to  exclude  peripheral  extraneous  light  by  means 
of  a  diaphragm  of  suitable  aperture,  and  it  is  even  possible  to  increase  the 
definition,  through  limiting  the  field  in  an  inferior  instrument  by  further 
reducing  the  size  of  this  aperture.  Thus  it  is  that  the  pin-hole  disk 
heightens  the  visual  acuteness  in  ametropes  who  view  objects  at  a  distance 
through  it.  While  the  same  proportionate  improvement  can  be  obtained 
in  a  similar  manner  at  finite  distance,  yet  it  would  be  exceedingly  diflScult 
to  accurately  place  the  pin-holes  before  the  pupils  of  both  eyes  for  reading 
binocularly.  To  obviate  this  impracticability,  while  still  securing  an  un- 
impaired field  of  fixation,  the  typoscope,  as  here  described,  seems  in  many 

183 


184  THJB    TYPOSCOrE. 

instances  to  effectively  serve  its  purpose.*     It  consists  of  a  rectangular 
plate  of  hard  rubber,  or  black  cardboard,  7  by  2]  inches,  provided  with  an 


One-half  of  Actual  Size. 


aperture  4|  by  |  inches,  centrally  located,  though  laterally  displaced  .so  as 
to  leave  sufficient  of  the  plate,  two  inches,  to  be  conveniently  held  between 
the  thumb  and  fingers,  when  it  is  placed  upon  the  book  or  paper,  and 
while  it  is  being  slid  down  over  the  column  in  reading.  The  central  aper- 
ture is  just  deep  enough  to  allow  two  lines  of  brevier  type  to  be  viewed  at 
a  time,  and  wide  enough  to  take  in  the  Avidth  of  an  average  column  of 
type,  as  shown  in  the  diagram.  The  author  has  found  it  to  be  especially 
serviceable  to  cataract  patients  and  amblyopes  wearing  high  corrections. 
The  former,  who  notably  suffer  greater  impairment  of  vision  from  ex- 
traneous light,  are  invariably  enabled  with  their  glasses  to  read  the  smallest 
type  by  the  aid  of  the  typoscope,  which  excludes  all  liglit  reflected  from 
the  ^rface  of  the  paper,  except  that  which  actually  affords  them  the 
necessary  contrast  between  it  and  the  type  within  tlie  slot.  The  device  is 
exceedingly  simple,  inexpensive,  and  easily  carried  in  the  pocket.  Its  utility 
is  easily  demonstrated  by  first  ascertaining  the  size  of  the  smallest  type 
which  the  patient  reads  witli  glasses,  and  then  allowing  the  patient  to  use 
the  typoscope  in  addition  to  them,  for  the  purpose  of  ascertaining  whether 
smaller  type  can  be  read,  or  not.  Even  in  the  latter  case  it  has  been  the 
authors  experience  that  patients  using  the  typoscope  claim  to  read  with 
less  sense  of  confusion. 


*  "I  am  delighted  with  the  typoscope.     It  rests  on  sound  physiological  principles,  and  will 
benefit  many  people." — li.  Knapp,   M.D.,   New  York. 


THE    CORRECTION    OF    DEPLETED 
DYNAMIC   REFRACTION. 

(PRESBYOPIA.) 

Rerised  reprint  from  the   "Optical  Journal,  "   June,   1895. 


To  render  this  subject  fully  comprehensive,  we  shall  first  briefly  describe 
Bonders'  Chart  of  the  Amplitudes  of  Accommodation  in  Emmetropia,  and 
in  which  the  "near  points"  are  represented  by  a  continuous  curved  line 
drawn  diagonally  through  its  field.  The  successive  ages,  between  10  and 
80  years,  are  therein  pointed  off  upon  the  uppermost  horizontal  line,  which 
also  represents  the  plane  of  the  eyes.  On  the  right  hand  margin,  the 
distances  of  the  near  points  from  the  eye  are  placed  opposite  to  the 
horizontal  lines  which  intersect  the  verticals  apportioned  to  the  various 
ages.  On  the  left  hand  margin  the  same  horizontals  are  numbered  in 
dioptrics  of  refraction,  coimted  from  the  zero-line  (o)  below,  which  is  sup- 
posed to  be  at  infinity.  The  latter  therefore  corresponds  to  the  refraction 
of  the  eye  when  at  rest — its  static  refraction.  Between  50  and  80  years  of 
age  it  will  be  noted  that  the  line  representing  the  static  refraction  curves 
slightly  downward  at  the  right,  showing  that  the  punctmn  remotum  becomes 
negative,  that  is  to  say,  the  refraction  of  the  emmetropic  eye  acquires 
hyperopia  from  age.  The  changes  in  the  refraction  of  the  feye  which  are 
accomplished  by  efforts  of  the  ciliary  muscle,  are  termed  its  dynamic 
refraction  or  power  of  accommodation.  The  amplitude  of  accommodation, 
at  any  age,  is  equal  to  the  number  of  ruled  spaces  between  the  zero-line 
and  the  curved  diagonal  line  above,  that  is  to  say,  equal  to  the  difference 
between  the  static  and  dynamic  refraction.    Therefore,  if  we  represent  the 

ransre  of  accommodation  by  a,  the  static  refraction  h\  r.  and  the  dynamic 

185 


186     THE  CORRECTION  OF  DEPLETED  DYNAMIC  EEFEACTION. 

refraction  by  p,  we  have 

a  =  p —  r 

as  the  formula  for  the  range  of  accommodation. 

In  emmctropia,  the  eye  is  adapted  to  infinity^,  so  that  the  static  refraction 
is  nil,  and  therefore 

a  =  p 

which  means  that  the  accommodative  effort  at  the  punctum  proximum  is 
equal  to  the  amplitude  of  accommodation. 

From  early  childhood  the  accommodation  is  shown  to  gradually  decrease, 
though  it  only  becomes  manifest  to  persons  as  they  approach  the  age  of  45, 
when  the  punctum  proximum  about  reaches  the  accustomed  distance  for 
reading.  The  patient  then  discovers  a  loss  of  distinctness  in  reading,  and 
experiences  a  desire  to  hold  his  book  at  a  slightly  greater  distance  than 
is  desirable.  This  serves  to  show  that  so  long  as  there  is  sufficient  power 
of  accommodation  in  reserve,  that  is  to  say,  capacity  to  adapt  the  eye  inside 
of  the  limit  of  the  usual  reading  distance,  there  will  be  no  asthenopic 
symptoms  from  what  is  commonly  called  Presbyopia,  if  by  the  latter  we 
only  mean  to  designate  a  recession  of  the  near  point  beyond  the  accustomed 
finite  occupation  distance.  Experience  has  also  shown  that  the  reserve 
accommodation  should  exist  in  a  more  or  less  definite  proportion  to  the 
amount  required  for  a  given  finite  distance.  It  would  prove  futile,  for 
instance,  to  prescribe  glasses  at  the  age  of  45,  which  would  artificially 
afford  the  patient  as  much  refraction  as  he  possessed  dynamically  at  the 
age  of  35. 

Landolf"  says  that  "the  accommodative  effort  is  not  to  be  measured  by 
/  ^ny  fixed  standard,  but  finds  its  expression  in  the  relation  between  the 

r\  ^  effoH  produced  and  the  entire  amount  of  accommodative  poirer  at  disposal. 

^^  It  follows  from  this  that  the  reserve  fund  of  accommodation  must  have  a 
relative,  and  not  an  absolute,  value;  it  must  be  a  quota  of  the  range  of 
accommodation."  He  has  found,  experimentally,  that  "a  continued  effort 
of  the  ciliary  muscle  is  practicable  only  when  it  calls  for  but  two-thirds,  or, 
at  the  utmost,  three-fourths  of  the  total  power  of  accommodation." 


*  The  Refraction  and  Accommodation  of  the  Eye,  page   339,   by  E.   Landolt,   M.D..   Paris; 
fa-anslated  by  C.  M.  Culver,  M.A.,  M.D.,  Philadelphia,   1886. 


TUE    COllKECTlOX    OF    DEPLETED    DYNAMIC    EEEKACTION.  l.S7 

In  the  accompanying  chart,  the  author  has  therefore  interposed  the  cor- 
Teetions  for  depleted  accommodation,  by  assnmin!]^  that  three-fourths  of  the 

STATIC   AND    DYNAMIC    REFRACTION 

IN    THE     EMMETROPIC    EYE    (Dondors) 
(CORRECTIONS    FOR     PRESBYOPIA    TO    25cm.     Prentice) 

AGES  : 

lo  IS      20   25   30   35   40   45   50   55   60   65   70   75   80 


14 
13 

12 
II 
10 

DIOPTRIPR- 

)T  PF 

OX.— 

X 

7-7 
8.3 

V^ 



9.1 

10. 

9 

8 

-^ 

V 





- 

II. I 
12.3 

7 



\ 

\ 

AC 

E: 

48 

53 

5S 

62 

14-3 

6 

\  CORRECTION:     i 

2 

2-7S 

35 

16.7 

5 
4 

\ 

\, 

1 
45 

1 

1 

1 
5,5 

1 
60 

3'- 

''     .t 

— C( 

">RRF< 

■^Tcri 

\ 

\ 

25. 

\ 

DISTANCE 

\ 

3 

\ 

40 

33-3 

2 

\ 

50. 

>^ 

.... 



571  ■ 

66 
So 

V, 

4- 

._ 

■^ 

.^^^.... 

■33 

.1 

"^ 

.j^OO 

■ 

""^ 

-— __ 

^^^ 

"^ 

— I 

i 

""'X^ 

2 

—AM 

=LITU 

DE-/! 

ccor 

IMOD 

ATior 

l:-f3 

5.-2 

1 
5— 1.75-1. 

25-075- o- 

25—0 

1          1       . 

3IA  • 

0  -"T.—o  c;       <^  7^0. 

75,  _-: 

3 

— 

C.    F.  PRENTICE,   DEL. ,  1887 


188  THE  CORRECTION  OF  DEPLETED  DYNAMIC  REFRACTION. 

total  accommodation  reijuired  at  25  cm.  is  used  in  reading  at  33J  cm. 
distance,  because  the  corrections  then  more  closely  correspond  to  Donders' 
table,  and  also  for  the  reason  that  the  author  has  found  it  more  satisfactory 
to  thus  calculate  in  practice.  For  instance,  the  emmetrope,  whose  ampli- 
tude of  accommodation  is  2.5  D.^  should  have  3  D.  of  refraction  to  read  at 
33^  cm.,  but,  as  he  must  also  have  one-third  of  the  accommodation 
required  at  this  distance  in  reserve,  he  should  have  4  D.  of  refraction  to 
read  comfortably.  Since  his  dynamic  refraction  is  2.5  D.  Ave  therefore 
give  him  -f  1.5  D.  glasses  to  supply  the  deficiency.  If  his  range  of 
accommodation  is  normal,  we  find,  by  the  chart,  that  he  should  be  50 
years  of  age.  In  this  manner  the  author  frequently  estimates  the  ages  of 
persons  with  surprising  accuracy. 

The  lenticular  corrections,  inserted  in  the  field  of  the  chart,  between  the 
ages  of  45  and  80  years,  added  to  the  corresponding  amplitude  of  accom- 
modation, less  the  acquired  hyperopia  noted  beneath,  are  found  in  each 
instance  to  amount  to  4  D.  Consequently,  in  any  case,  the  requisite  3  D. 
to  read  at  %o\  cm.  represents  three-fourths  of  the  total  power  supplied 
both  dynamically  and  artificially,  by  lenses. 

Of  course,  the  corrections  in  the  table,  are  only  applicable  in  such  cases 
where  the  reading  distance  is  no  greater  than  33 J  cm. ,  and  wherein  the 
amplitude  of  accommodation  is  found  to  correspond  to  Donders'  deter- 
minations. 

The  chief  value  of  the  chart  therefore  exists  only  in  the  fact  that  it  serves 
to  show,  by  comparison  with  any  given  case  under  examination,  to  what 
extent  its  amplitude  of  accommodation  differs  from  the  accepted  normal 
state  as  determined  by  Donders.  This  cannot  be  too  highly  estimated, 
however,  for  Landolt  *  says  :  ' '  Donders'  diagram  corresponds  so  perfectly 
to  the  natural  condition  of  things,  that,  in  every  case  where  the  amplitude 
of  accommodation  is  less  than  is  indicated  thereon,  we  may  safely  diagnose 
a  weakness  of  accommodation,  and,  in  case  of  any  considerable  difference, 
we  may  admit  a  paresis  of  this  function."  A  knowledge  of  the  patient's 
condition  of  health  and  habits  will  of  course  assist  greatly  in  arriving 
at  a  definite  decision. 

The  piindinn  proximum  is  tliat  point  located  at  finite  distance,  at  which 

*LandoU's  work,  rR§e  ^^^- 


THE    CORKECTIOM    OK    DEIVLETED    DYMAiMIC    REFKACTION. 


189 


the  patient  is  still  able  to  distinctly  sec  small  printecl  characters,  such 
as  diamond  type,  dots,  or  fine  lines.  The  size  of 
the  test-objects  should,  of  course,  be  in  proportion 
to  the  visual  acuteness  of  the  eyes,  since  there  are 
])ersons,  who,  though  possessing  a  good  range  of 
accommodation,  cannot  read  fine  print  at  any  dis- 
tance, simply  because  their  visual  acuteness  is  in- 
sufficient. Therefore,  in  cases  where  the  visual  acute- 
ness is  -y-  ,  or  less,  it  is  preferable  to  use  more 
heavily  executed  test-objects.  For  a  normal  visual 
acuteness,  the  author  finds  it  convenient  to  use  the 
characteis  liere  shown  and  which  are  engraved  on  a 
circular  disk  mounted  in  a  handle,  to  be  held  by  the 
patient.  On  flic  obverse  side  of  the  disk,  the  same, 
though  more  heavily  drawn  figure,  a,  is  used  when 
the  visual  acuteness  is  subnormal. 

To  determine  the  position  of  the  pimctum  proxi- 
muni,  the  patient  is  requested  to  binocularly  fix  the 
central  dot  of  the  test-object  at  about  his  usual  read- 
ing distance,  and  to  gradually  draw  it  nearer  to  the 
eyes  until  it  just  begins  to  blur;  the  latter  effect 
being  made  more  noticeable  to  the  patient  through  the  tendency  of  the  dot 
to  fuse  with  the  lines  of  the  square  which  encloses  it.  The  punctum  proxi- 
mum  is  reached  the  instant  before  the  blurring  is  observed.  This  experi- 
ment should  be  repeated  several  times,  and  it  is  generally  also  advisable 
to  verify  it  by  moving  the  dot  slightly  nearer  than  the  punctum  proxi- 
mum,  having  the  patient  fix  the  dot  attentively  while  it  is  gradually 
withdrawn  to  the  position  where  it  again  appears  sharply  defin^  By 
using  a  tape,  Avhich  is  graduated  to  dioptrics  of  refraction,  we  may  read 
directly  from  its  graduations,  the  amplitudes  of  accommodation  in  emme- 
tropia,  provided  the  distance  is  measured  from  the  cornea  to  the  test-object, 
on  the  median  line.  The  same  procedure  will  apply  in  any  case  of 
ametropia,  when  the  distance  glasses  which  correct  it  are  worn  at  the  time 
the  above  measurement  is  made.  In  the  latter  case,  calculation  will  of 
course  be  greatly  simplified. 

To  those  familiar  with  the  art  of  fitting,  glasses  it  is,  in  most  instances, 


190     THE  COEEECTION  OF  DEPLETED  DYNAMIC  REFEACTION. 

comparatively  easy  to  determine  the  proper  distance  glasses,  as  well  as  for 
them  to  predict,  with  reasonable  certainty,  that  the  lenses  will  be  worn, 
with  comfort.  But  when  the  case  is  complicated  with  a  loss  of  dynamic 
refraction,  and  unless  extreme  caution  is  used,  there  is  great  danger  in 
giving  the  patient  an  over-correction,  thus  ultimately  making  a  change  in 
the  prescribed  reading  glasses  necessary. 

It  is  therefore  of  great  importance  to  the  optical  practitioner,  particularly 
if  the  cost  of  a  subsequent  change  in  the  reading  glasses  is  to  be  borne  by 
him,  that  he  should  be  able  to  predict  the  correctness  of  the  reading  glasses 
with  the  same  degree  of  certainty  that  he  feels  in  respect  to  the  glasses 
which  he  prescribes  for  distance.  In  prescribing  for  depleted  accommoda- 
tion there  are  at  present  two  methods  in  vogue : 

1.  The  impirical  method;  that  of  prescribing  the  glasses  which  have 
been  accorded  to  a  given  age  by  Bonders  in  his  table. 

2.  The  physical  method;  that  of  locating  the  punctum  proximum  in  each 
individual  case,  and  using  this  as  the  basis  for  calculating  the  loss  of  ac- 
commodation which  is  to  be  compensated  for. 

The  latter  is  the  only  accurate  and  reliable  method,  j^et  even  in  apply- 
ing it  we  are  frequently  hampered  by  the  patient's  indecision  as  to  the  dis- 
tance at  which  he  habitually  performs  his  near  work.  In  the  practitioner's 
office  the  patient  may  indicate  the  distance  as  being  thirteen  inches,  whereas 
at  his  own  occupation  he  perhaps  finds  that  it  is  twenty.  To  avoid  this 
feature  of  uncertainty  as  much  as  possible,  it  is  consequently  prudent  to 
have  the  patient  state  the  nature  of  his  occupation,  and  to  have  him  assume 
his  accustomed  position  of  the  head,  arms  and  body  when  engaged  in  near 
work.  Then  measure  the  distance  from  the  eye,  during  convergence  to 
the  median  line,  by  the  dioptral  tape,  and  note  it  as  the  desired  reading 
distance,  which  is  after  all  the  real  and  only  distance  to  be  considered  in 
the  calculation  for  reading  glasses. 

As  an  example,  let  us  take  an  emmetrope  who  has  a  rang^  of  accommo- 
dation of  2  D.,  and  who  indicates,  by  the  aforesaid  measurement,  that  he 
desires  to  see  at  the  distance  which  corresponds  to  a  refraction  of  2.35  D. 
by  the  tape.  As  this  amount  should  represent  only  three-fourths  of  the 
total  refraction,  to  allow  him  one-fourth  in  reserve,  it  follows  that  the  total 


THE  CORRECTION  OP  DEPLETED  DYNAMIC  REFRACTION.  191 

refraction  should  be  3  D.     As  he  is  capable  of  contributing  2  D. ,  dynami- 
cally, we  give  him  -\-  ^  D.  glasses  to  supply  the  deficiency  for  reading. 

Therefore,  to  ascertain  the  reading  glasses  for  any  given  case  of  emme- 
tropia^  we  have  only  to  follow  the  simple  rule  : 

Increase  the  refraction  corresponding  to  the  desired  reading 
distance  by  one-third,  and  subtract  therefrom  the  patient's 
amphtude  of  accommodation. 

It  happens  occasionally,  as  in  the  above  example,  that  the  punctum 
proximum  is  too  far  distant  to  enable  the  patient  to  see  the  test-object  dis- 
tinctly. In  such  cases  it  is  convenient  to  assist  the  patient  by  a  lens  which 
will  enable  him  to  do  so.  Let  us  suppose  that  we  have  assisted  the  patient 
impirically  by  a  1  Z?.  lens,  when  he  will,  by  the  use  of  his  2  D.  of  accom- 
modation, be  able  to  see  the  test-object  distinctly  at  the  3  D.  distance.  By 
deducting  the  1  D.  lens  we  then  find  his  amplitude  of  accommodation, 
which  is  2  Z>. ,  and  proceed  by  the  rule  given. 

In  ametropia^^  provided  it  is  corrected  by  the  distance  glasses,  which 
then  virtually  render  the  patient  emmetropic  in  accommodation,  we  pro- 
ceed by  the  same  rule.  The  full  reading  correction  will  in  this  case  be 
equal  to  the  amount  found  by  the  rule,  plus  the  distance  correction. 
Should  the  accommodation  be  insufl5cient  to  definitely  locate  the  punctum 
proximum,  the  patient,  here  too,  is  to  be  assisted  by  a  lens  which  will 
likewise  have  to  be  deducted  to  ascertain  the  amplitude  of  accommodation. 
An  exception  to  an  application  of  the  rule  is  found  in  those  cases  of 
myopia  where  the  punctum  proximum  is  nearer  than  the  desired  reading 
distance.  In  such  cases  it  is  customary  to  deduct  the  refraction  corre- 
sponding to  the  desired  reading  distance  from  the  distance  correction.  In 
some  instances,  where  the  amplitude  of  accommodation  is  considerable. 


*  When  the  ametropia  is  not  corrected,  that  is  to  say,  when  the  distance  glasses  are  not  worn  during 
the  measurement  of  the  range  of  accommodation  we  must  resort  to  the  formula  :  a^p  —  r.  In  hyper- 
opia the  punctum  remotum  is  behind  the  eye,  therefore  the  refraction  is  negative,  so  that 

a=p  —  ( —  r)=p  -\-  r 
which  means  that  the  range  of  accommodation  is  equal  to  the  refraction  at  the  punctum  proximum, 
plus  the  refraction  of  the  lens  which  corrects  the  hyperopia. 

In  myopia  the  amplitude  of  accommodation  is  equal  to  the  difference  between  the  refraction  at  the 
near  point,  and  the  refraction  of  the  lens  which  corrects  the  myopia.  For  a  more  exhaustive  discussion 
of  this  subject,  the  reader  is  referred  to  Dr.  Landolt's  work,  in  which  the  physical  portion  is  treated  at 
greater  length  and  more  lucidly  than  in  any  other  medical  publication  which  has  come  to  the  author'a 
notice. 


102 


THE    COERECTION    OF   DEPLETED   DYNAMIC    BEFBACTIOlf. 


and  more  especially  in  young  persons,  the  patient  will  prefer  to  use  his 
distance  correction  for  all  purposes. 

From  this  discussion  it  must  be  evident  to  anyone  proficient  in  the 
practice  of  optometry  that,  as  a  matter  of  fact,  greater  skill  and  knowledge 
is  required  to  scientifically  determine  the  proper  glasses  for  reading,  com- 
monly known  as  presbyopic  corrections,  than  is  needed  to  ascertain  the 
lenticular  requirements  for  distance.  Nevertheless,  some  oculists  have  ex- 
pressed the  opinion  that  opticians  should  only  be  permitted  to  adapt  glasses 
for  presbyopee.  If  the  same  medical  gentlemen  were  better  informed  ia 
optics,  they  would  imdoubtedly  deny  opticians  all  rights  in  the  matter. 


CLINICS  IN  OPTOMETRY 

By  C.  H.  Brown,  M.D. 

Graduat<>  Uniyersity  of  Pennsylvania  ;  Professor  of  Optics  and  ftefraction  ;  formerly  Phypiciaa 

in  Philadelphia  Hospital ;  Member  of  Philadelphia  C-onnty,  Pennsylvania 

Stat*  and  American  Medical  Societies 


"Clinics    in    Optometry"    is   a 
unique  work  in  the  field  of  practical 
refraction  and  fills  a  want  that  has 
^^^^^^^^^  been    seriously  felt  both  by  oculists 

l^lTn^^BHI  '^'^^  optometrists. 

^^^^^^^^^^^  The   book    is   a  compilation  of 

ppTnTIi^LMBIj  optometric  clinics,  each  clinic  being 

complete  in  itself.  Together  they 
cover  all  manner  of  refractive  eye 
defects,  from  the  simplest  to  the 
most  complicated,  giving  in  minutest 
detail  the  proper  procedure  to  follow 
in  the  diagnosis,  treatment  and  correction  of  all  such  defects. 

No  case  can  come  before  you  that  you  cannot  find  a  similar 
case  thoroughly  explained  in  all  its  phases  in  this  useful  volume, 
making  mistakes  or  oversights  impossible  and  assuring  correct  and 
successful  treatment. 

The  author's  experience  in  teaching  the  science  of  refraction 
to  thousands  of  pupils  peculiarly  equipped  him  for  compiling  these 
clinics,  all  of  which  are  actual  cases  of  refractive  error  that  came 
before  him  in  his  practice  as  an  oculist. 

A  copious  index  makes  reference  to  any  particular  case,  test 
or  method,  the  work  of  a  moment. 


Sent  postpaid  on  receipt  of  $i.50  ies,  3d.) 


PUBLISHKD   BY 


The  Keystone  Publishing  Co. 

809-81  i-Si.^  North  19x11  Street,  Philadelphia,  U.S.A. 


THE   OPTICIAN'S    MANUAL 

VOL.  I. 


By  C.  H.  Brown,  M.  D. 

Graduate    University   of   Pennsylvania;    Professor   of   Optics    and    Refraction;    formerly 

Physician   in   Philadelphia   Hospital;    Member   of   Philadelphia   County, 

Pennsylvania    State    and    American    Medical    Societies. 


The 

*^' ;  OPTICIANS 
>!'     MANUAL 


The  Optician's  Manual,  Vol.  I,  was 
the  most  popular  and  useful  work  on 
practical  refraction  ever  written,  and  has 
been  the  entire  optical  education  of 
many  hundred  successful  refractionists. 
The  knowledg"e  it  contains  was  more  ef- 
fective in  building-  up  the  optical  profes- 
sion than  any  other  educational  factor. 
It  is,  in  fact,  the  foundation  structure  of 
all  optical  knowledge  as  the  titles  of  its 
ten  chapters  show: 

-Introductory  Remarks. 

-The  Eye  Anatomically. 

-The  Eye  Optically;  or.  The  Physiology  of  Vision. 

-Optics. 

-Lenses. 

-Numbering  of  Lenses. 

-The  Use  and  Value  of  Glasses. 

-Outfit  Required. 

-Method  of  Examination. 

-Presbyopia. 

The  Optician's  Manual,  Vol.  I,  was  the  first  important 
treatise  published  on  eye  refraction  and  spectacle  fitting.  It 
is  the  recognized  standard  text-book  on  practical  refraction, 
being  used  as  such  in  all  schools  of  Optics.  A  study  of  it  is 
essential  to  an  intelligent  appreciation  of  its  companion  treatise, 
The  Optician's  Manual,  Vol.  II,  described  on  the  opposite 
page.  A  comprehensive  index  adds  much  to  its  usefulness  to 
both  student  and  practitioner. 

Bound  in  Cloth — 422  pages — colored  plates  and  illustrations. 
Sent  postpaid  on  receipt  of  $1.50  ^63,  3d.) 


Chapter 

L- 

Chapter 

H. 

Chapter 

III. 

Chapter 

IV. 

Chapter 

V.- 

Chapter 

VI. 

Chapter 

VII.- 

Chapter 

VIII.- 

Chapter 

IX.- 

Chapter 

X.- 

Published  by 

T^HS  Keystone  Publishing   Co. 
809-811-813  North  19TH  Street,  Philadelphia,  U.  S.  A. 


THE  OPTICIAN'S  MANUAL 

VOL.   11. 


By  C.  H.  Brown,  M.  D. 

Oraduate    University    of    Pennsylvania;    Professor   of   Optics    and    Refraction;    formerly 

Physician   in   Philadelphia   Hospital;    Member   of   Philadelphia  County, 

Pennsylvania  State  and   American  Medical   Societies. 


The  Optician's  Manual,  Vol.  II.,  is 
a  direct  continuation  of  The  Optician's 
Manual,  Vol.  I.,  being-  a  much  more 
advanced  and  comprehensive  treatise. 
It  covers  in  minutest  detail  the  four 
great  subdivisions  of  practical  eye  re- 
fraction, viz : 

Myopia. 
Hypermetropia. 
Astigmatism. 
Muscular  Anomalies. 


It  contains  the  most  authoritative  and  complete  re- 
searches up  to  date  on  these  subjects,  treated  by  the  master 
hand  of  an  eminent  oculist  and  optical  teacher.  It  is  thor- 
oughly practical,  explicit  in  statement  and  accurate  as  to  fact. 
All  refractive  errors  and  complications  are  clearly  explained, 
and  the  methods  of  correction  thoroughly  elucidated. 

This  book  fills  the  last  great  vi'ant  in  higher  refractive 
optics,  and  the  knowledge  contained  in  it  marks  the  standard 
of  professionalism. 

Bound  in  Cloth — 408  pages — with  illustrations. 
Sent  postpaid  on  receipt  of  $1.50  (6s.  3d.) 


Published  by 

Thk   Keystone   Publishing  Co. 
S09-811-813  North  19TH  Street,  Philadelphia,  U.S.  A. 


THE 

PRINCIPLES  of  REFRACTION 

in  the  Human  Eye,  Based  on  the  Laws  of 
Conjugate  Foci 


By  Swan  M.  Burnett,  M.  D.,  Ph.  D. 

Formerly  Professor  of  Ophthalmology  and  Otology  in  the  Georgetown   University 
Medical  School;  Director  of  the  Eye  and  Ear  Clinic,  Central  Dispensary 
and  Emergency  Hospital;  Ophthalmologist  to  the  Children's  Hos- 
pital and  to  Providence  Hospital,  etc.,  Washington,  D.  C. 


In  this  treatise  the  student  is  given  a  condensed  but  thor- 
ough grounding  in  the  principles  of  refraction  according  to  a 
method  which  is  both  easy  and  fundamental.  The  few  laws 
governing  the  conjugate  foci  lie  at  the  basis  of  whatever  per- 
tains to  the  relations  of  the  o1)ject  and  its  image. 

To  bring  all  the  phenomena  manifest  in  the  refraction  of 
the  human  eye  consecutively  under  a  common  explanation  by 
these  simple  laws  is,  we  believe,  here  undertaken  for  the  first 
time.  The  comprehension  of  much  which  has  hitherto  seemed 
difficult  to  the  average  student  has  thus  been  rendered  much 
easier.  This  is  especially  true  of  the  theory  of  Skiascopy, 
which  is  here  elucidated  in  a  manner  much  more  simple  and 
direct  than  by  any  method  hitherto  offered. 

The  authorship  is  sufficient  assurance  of  the  thoroughness 
of  the  work.  Dr.  Burnett  was  recognized  as  one  of  the  great- 
est authorities  on  eye  refraction,  and  this  treatise  may  be 
described  as  the  crystallization  of  his  life-work  in  this  field. 

The  text  is  elucidated  by  24  original  diagrams,  which 
were  executed  by  Qias.  F.  Prentice,  M.E.,  whose  pre-emi- 
nence in  mathematical  optics  is  recognized  by  all  ophthalmol- 
ogists. 

Bound  in  Siik  Cloth. 

Sent  postpaid  to  any  part  of  the  world  on  receipt  of  price, 

$1.00  (4s.  2d.) 


Published  by 
Thk  Keystone  Pubi^ishixg  Co. 
809-811-813  North  19TH  Street,  Philadelphia,  U.  S, 


PHYSIOLOGIC  OPTICS 

Ocular  Dioptrics — Functions    of    the  Retina — Ocular 
Movements  and  Binocular  Vision 


By  Df.  M.  Tscheming 

IHrector  of   tlie   Lalioratory  of  Ophthalmology  at  the   Sorbonne,    P&ris. 


AUTHORIZED   TRANSLATION 
By  Carl  Wciland,  M.  D. 

Fomer  Chief  of  Clinic  in  the  Eye  Department  of  the  Jefferson  College  Uospitai, 
Philadelphia,  Pa. 


This  book  is  recognized  in  the  scientific  and  medical 
world  as  the  one  complete  and  authoritative  treatise  on 
physiologic  optics.  Its  distinguished  author  is  admittedly 
the  greatest  authority  on  this  subject,  and  his  book  embodies 
not  only  his  own  researches,  but  those  of  the  several  hundred 
investigators  who,  in  the  past  hundred  years,  made  the  eye 
their  specialty  and  life  study. 

Tscherning  has  sifted  the  gold  of  all  optical  research  from 
the  dross,  and  his  book,  as  now  published  in  English,  with 
many  additions,  is  the  most  valuable  mine  of  reliable  optical 
knowledge  within  reach  of  ophthalmologists.  It  contains  380 
pages  and  212  illustrations,  and  its  reference  list  comprises  the 
entire  galaxy  of  scientists  who  have  made  the  century  famous 
in  the  world  of  optics. 

The  chapters  on  Ophthalmometry,  Ophthalmoscopy,  Ac- 
commodation, Astigmatism,  Aberration  and  Entoptic  Phenom- 
ena, etc. — in  fact,  the  entire  book  contains  so  much  that  is  new, 
practical  and  necessary  that  no  refractionist  can  afford  to  be 
without  it. 

Bound  in  Cloth.     380  Pages,  212  lilustratioas. 
Price  $2.50  (IDs.  5d.) 


Published  by 

The   Keystone   Publishing  Co. 

809-81 1-813  North  19TH  Street,  Philadelphia,  U.  S.  A. 


SKIASCOPY 

AND  THE  USE  OF  THE  RETINOSCOPE 


By  Geo.  A.  Rogers 

rormci'iy    I'rofessor    in    the    Northern   Illinois   College    of   Ophthalmology   and    Otology, 

Cliii'afro;    Principal    of    the    Chicago    Post-Graduate    College    of 

Optometry;    Lecturer    and    Specialist    on 

Scientific   Eye   Refraction. 


■^B^ 


A  Treatise  on  the  Shadow  Test 
in  its  Practical  Application  to  the 
Work  of  Refraction,  with  an  Ex= 
planation  in  Detail  of  the  Optical 
Principles  on  which  the  Science 
is  Based. 


This  work  far  excels  all  previous  treatises  on  the  sub- 
ject in  comprehensiveness  and  practical  value  to  the  refrac- 
tionist.  It  not  only  explains  the  test,  but  expounds  fully  and 
explicitly  the  principles  underlying  it — not  only  the  phe- 
nomena revealed  by  the  test,  but  the  why  and  wherefore  of 
such  phenomena. 

It  contains  a  full  description  of  skiascopic  apparatus,  in- 
cluding the  latest  and  most  approved  instruments. 

In  depth  of  research,  wealth  of  illustration  and  scientific 
completeness  this  work  is  unique. 

Bound  in  cloth;  contains  231  pages  and  73  illustrations 
and  colored  plates. 

Sent  postpaid  to  any  part  of    the    world    on    receipt    of 
$1.00  (4s.  2d.) 


Published  by 
The  Keystonk  Publishing  Co. 
-S11-813  North  19TH  Street,  Philadelphia,  U.  S.  A. 


TESTS  AND  STUDIES 

OF  THE 

OCULAR  MUSCLES 


By  Ernest  E.  Maddox,  M.  D.,  F.  R.  C.  S.,  Ed. 

Ophthalmic  Surgeon  to  the  Royal  Victoria  Hospital,   Bournemouth,   England;   formerly 
Sj-me    Surgical    Fellow,    Edinburgh    University. 


|j  Tests  and  Studies 

j!  of  the 

;    OGular  Muscle|, 


This  book  is  universally  recog- 
nized as  the  standard  treatise  on 
the  muscles  of  the  eye,  their  func- 
tions, anomalies,  insufficiencies, 
tests  and  optical  treatment. 

All  opticians  recognize  that  the 
subdivision  of  refractive  work  that 
is    most    troublesome    is    muscular 
anomalies.     Even  those  who  have 
mastered  all  the  other  intricacies  ot 
visual  correction  will  often  find  their 
skill    frustrated    and    their    efforts 
nullified  if  they  have  not  thoroughly 
mastered  the  ocular  muscles. 
The  eye  specialist  can  thoroughly  equip  himself  in  this 
fundamental  essential  by  studying  the  work  of  Dr.  Maddox 
who  is  known  in  the  world  of  medicine  as  the  greatest  in- 
vestigator and  authority  on  the  subject  of  eye  muscles. 

The  present  volume  is  the  second  edition  of  the  work, 
specially  revised  and  enlarged  by  the  author.  It  is  copiously 
illustrated  and  the  comprehensive  index  greatly  facilitates 
reference. 

Bound  in  silk  cloth — 261   pages — no  illustrations. 
Sent  postpaid  on  receipt  of  price  $1,50  (6s.  3d.) 


Published  by 

The  Keystone   Publishing  Co. 
809-81 1-813  North  19TH  Street,  Philadelphia,  U.  S. 


Optometric  Record  Book 


A  record-book,  wherein  to  record  optometric  examina- 
tions, is  an  indispensable  adjunct  of  an  optician's  outfit. 

The  Keystone  Optometric  Record-book  was  specially  pre- 
pared for  this  purpose.  It  excels  all  others  in  being-  not  only  a 
record-book,  but  an  invaluable  guide  in  examination. 

The  book  contains  two  hundred  record  forms  with  printed 
headings,  sug-gesting,  in  the  proper  order,  the  course  of  ex- 
amination that  should  be  pursued  to  obtain  most  accurate  re- 
sults. 

Each  book  has  an  index,  which  enables  the  optician  to 
refer  instantly  to  the  case  of  any  particular  patient. 

The  Keystone  Record-book  diminishes  the  time  and 
labor  required  for  examinations,  obviates  possible  oversights 
from  carelessness,  and  assures  a  systematic  and  thorough  ex- 
amination of  the  eye,  as  well  as  furnishes  a  permanent  record 
of  all  examinations. 

Sent  postpaid  on  receipt  of  $1.00  (4s.  2d.) 


Published  by 

The   Keystone  Publishing  Co. 

809-811-813  North  19TH  Street,  Philadelphia,  U.  S.  A. 


TcK     ?riOMETRY  LIBRARY 

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"HOiVlE  USE'  ' 


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